The main features of Christian monastic life are prayers (frequently chanted) and scholarship (reading, writing, healing etc.), sometimes corporeal penance for some Orthodox monks (e.g. flagellants, heavy fasting, etc.).
The main features of Buddhist monks from Shaolin are chanting, meditation and kick-ass martial arts prowess (which they originally learned from retired soldiers who became monks).
For some reason, fiction authors sometimes can't see the difference.
It's not uncommon (usually in fantasy settings) to see a Christian or quasi-Christian monk (often complete with robe, hood, tonsure, Christian name, etc.) whose main defining skills are one or all of the following:
Bare-hand (and bare-foot) fighting
Shaolin-style fighting with staves, nunchakus etc.
Most of the major characters in Alive: The Final Evolution are given superpowers and pushed to the Despair Event Horizon by being quasi-possessed by suicidal energy beings from space and not actually committing suicide; a significant subset of these are brainwashed by one of the others, and become the main antagonists of the first half. The nicest of these is a really sweet Japanese Catholic priest who for reasons that are never even slightly addressed has probably the most killer karate in the series. His power is to turn people to stone. He doesn't use it much, but he kicks a lot of ass.
In the climactic battle of the first movement, Katsumata has him constantly fighting on the losing side, whichever that happens to be, because he's powerful, his doing this is both the most plausible and the easiest to arrange since he's the type to root for the underdog anyway, and the true evil goal requires as much combat and hatred and use of psychic powers as possible in the location where he's arranged for the battle to be staged so they can awaken the powerful thing in the lake.
Do regular clergymen count? We hope so, because Father MacGruder from Brain Dead will let you know he kicks ass for the Lord!
Friar Tuck in many Robin Hood films reflects his rather martial nature described in the original stories.
Jackie Chan's kung-fu adventure Armor Of God has a cult of evil Western monks he must fight at the end. Of course, this being Jackie, everyone knows kung-fu, but...
French movie Les Rivières Pourpres 2 features a monk (complete with outfit) committing murders, knowing martial arts, and doing parkour. He was played by Cyril Rafaelli, one of the leading parkour experts
The Gamers: Dorkness Rising is, in part, about a D&D campaign whose players are in a somewhat antagonistic relationship with their GM. One player insists on playing a Kung-Fu Monk, which he says that he can do because it's a base D&D character class. The GM complains that it can't work because the campaign takes place in a traditional European fantasy setting, but relents, on the condition that he play the character as being a Western monk who also has the character class's Kung-Fu skills.
The 1998 incredibly low budget (verging on home made) Irish action film Fatal Deviation features a bunch of homeless monks in the real town of Trim helping the hero Jimmy Bennet (played by Jimmy Bennet) train for an ancient tournament in which he must defeat a member of the local "Drug Lords Gang" in order to break their power. Or something. The monks really have nothing to do with the plot.
Brother Cadfael, of the eponymous series by Ellis Peters, is a 12th century Benedictine herbalist. However, he used to be a sailor and soldier, having participated in the First Crusade, and can still kick ass if required. It's made clear that Brother Cadfael's fighting skills are justified as part of his backstory rather than a standard skill for all medieval monks.
Friar Tuck of the Robin Hood stories is usually depicted as a capable warrior. In the ballad recounting Robin's encounter with "the curtal friar," we are told explicitly: "The friar had on a harness good, / And on his head a cap of steel, / Broad sword and buckler by his side, / And they became him well." Some versions of the character suggest that Tuck had soldier training, which would have given him experience with various medieval weapons as well as unarmed combat.
The Abellican Monks from R. A. Salvatore's Demon Wars series are a Crystal Dragon Jesus religion with very strong martial traditions (one of them, Marcallo De'Unnero, was considered one of the top hand-to-hand combatants in the world). However, not all of the monks follow this path, only those who actually plan on serving as holy warriors. The majority are fairly typical western priest/scholars (with perhaps a bit of Sinister Minister thrown in, depending on whether the Abellican Church is currently corrupt or not). Oh except there's their magical gemstones, which have a different ability depending on what kind of gemstone it is (rubies shoot fireballs, hematites heal, and various other things). Marcallo for example uses a tigerseye to grow tiger claws.
Redwall might be a borderline example, as the superpowered Abbey Warrior usually starts out as a novice in the Abbey and quickly becomes awesome at swordplay, and even the ones who aren't warriors tend to be able to hold their own in a fight.
More books don't star an Abbey native than do, by some way, and the abbey is also notably lacking in any actual religion whatsoever. And most of the order don't generally know any combat skills; when Abbey natives do, it's generally because they retired there from more warlike lifestyles, like blind Cregga Rose-Eyes, who used to be a giant berserker and pulls this out to only slightly less devastating effect when her home comes under attack. Why beasts keep attacking the only thing resembling a fortress in leagues when it has no significant wealth to justify the investiture of effort...
Well, some of them want a nice shiny fortress of their own.
Averted in Neal Stephenson's Anathem, where there is one concent (i.e. monastery) of warrior monks who in turn inspire non-mathic (i.e. popular) entertainment, but they're the only one like it in the world.
German TV-show Lasko - Die Faust Gottes (i.e. Lasko - The Fist of God) is the embodiment of this trope. It's about a soldier who turned into a monk after some backstory in Kosovo. He now works for the secret order Pungus Dei to help people in distress. It all started as a regular movie that even got released in UK under the name Lasko Death Train and later turned into a series.
You know it doesn't work when you see those robes flying around... they're just not made for kung fu. And a heavy, rare MALE instance of Magic Skirt!
It helps that most of the ads for the series have him without the robes.
Obviously: David Carradine in Kung Fu. While justified in that he was a Shaolin monk, this is what probably started the entire trope of "all monks knowing kung-fu" by generalizing from this original series.
Shows up in the Doctor Who episode "Tooth and Claw". When the monks announce they're taking over a mansion, they reveal orange robes under their cloaks and do a dramatic, slow-motion leap over the heads of the astonished housekeepers. Normally orange robes are worn by Buddhists, while frocks were worn by Franciscan friars - which are quite different from monks. Neither Buddhists nor Franciscans should have had a monastery in 19th century Scotland.
Advanced Dungeons and Dragons (AD&D) 1E included a class called the Monk, inspired by the Kung-Fu monk. Many players, and not a few adventure writers, however, made them out to be some form of Western monastic. This version was in print by 1978. As an interesting note, AD&D 2E made the fighting monk a "kit" (modification) of the cleric, and deleted the monk as a class of its own.
4e averted this when they made it abundantly and explicitly clear that the revived "Monk" class was of Shaolin derivation: they are typically depicted in artwork as wearing gi-like robes, hair up in a topknot, and bare-chested if male. Fluff material often describes them (or their styles) coming "from the east" or wherever the stock Asia-equivalent is. Finally, ALL their moves sound like standard Wu Xia fair (Spinning Leopard Maneuver, Dancing Cobra, Strike the Golden Bell, Centered Breath Style, etc.) It doesn't stop some players from making the mistake though, ala the Dorkness Rising example above.
All monks know hand-to-hand combat and possess supernatural powers in Legend of the Five Rings. This isn't so bad, since it's explicitly a combination of various Eastern cultures, but Shaolin-style monks outnumbering Zen-style monks comes straight from pop culture.
The whole point of becoming an Immaculate Monk in Exalted is to reach enlightened understanding of Essence... So that you could learn Celestial Martial Arts and kick asses of rowdy gods.
Heroes of Might and Magic 3: Monks from the Western European-style Castle are dressed in hooded robes and... throw fireballs. Their upgraded version, Zealots, also have no penalty in hand-to-hand combat.
Somewhat complicated by the fact that the most direct influence on Heroes 3's Monks were... Heroes 2's Druids, who were pretty much the exact same unit (including clothing), only with another name and in another castle.
Dungeon Keeper 2: The Monk there looks like a stereotypical Catholic monk: fat, round-faced, with a tonsured head and dressed in a brown robe. He is a skilled bare-hand fighter.
Somewhat inverted in the English dub of Dynasty Warriors: Taoist monk Zhang Jiao is characterized as a televangelist.
Ragnarok Online has a Monk class which seems to be a cross between Catholic priests and Shaolin monks.
Might and Magic 7 has a Monk class that is a combination of the Shaolin and Western style of monk, with magic for good measure.
In-story, this is true for the Blades, an order of knights sworn to protect the Emperor, although it's less kung fu and more swordfighting. Allegedly, most of them are monks in their spare time. But especially Brother Jauffere, who actually is wearing a habit and working in a priory when you meet him. While carrying a sword on his back.
The Blades take on the roles of spies and bodyguards, essentially being His Imperial Majesties Secret Service. Their being monks generally comes from being undercover or retired. So the Blades are spies first, knights second, monks last.
This has actually been the case for every Elder Scrolls game. Somewhat justified, as most religions in Tamriel usually require someone getting the shit beaten out of them. Most notable are the Ordinators and the Order of Talos (who often double as Blades).
The Monk class in Final Fantasy games is usually an aversion, being almost universally Asian-styled. Even when there's no obviously Asian culture from which it might hail.
Although in Final Fantasy XI, there's a "far east" which, although we never see it, is clearly implied to be a thinly-veiled Japan analogue. It's also the origin for the Monk, Samurai, and Ninja techniques (although for some reason, the medieval French nation of San d'Oria picked up the Monk training and adapted it for their own use). There's even a neat in-story justification for why, of all the beastman races, only the Yagudo have Samurais?they have good relations with the far east.
Diablo III monks have an eastern orthodox appearance and speak in an eastern European accent, but use attacks a a fighting style more similar to stereotypical East Asian martial arts. This is less a case of not doing the research, though, and more a case of deliberately combining things in an unusual way.
Besides, have you seen the beards of an Eastern Orthodox priest or monk? Those things should require a permit, they're so badass!
In World of Warcraft there are various different types of enemies inside the Scarlet Monastery. The scarlet monks? They fight bare-fisted, wear headbands and kick spellcasters to interrupt them.
The Mists of Pandaria expansion introduces shaolin-style monks as a new character class. This is justified by the simultaneous introduction of the pandaren as a new playable race, obviously based heavily on east-Asian cultures.
In Ancient Domains of Mystery, the Monk class is not just a class with strong unarmed combat bonuses, it is in fact completely unrelated to religion, even though religion plays a significant role in the game. The only classes with religion-related bonuses and abilities are Priests and Paladins; Monks are exactly as inherently religious as any other melee class.
Lunar: The Silver Star makes the party White Mage, apprentice priestess Jessica, the third strongest character in the game, and she can fight with her fists. Her strength is underutilized in the last third of the game however, in that enemies hit so hard she must be kept on constant healing duty.
In Time Commando, you fight these in Feudal Japan and Medieval Europe.
In Touhou when it came time to assign fighting styles in the fighting game spin-off Hopeless Masquerade, of course the Buddhist monks of the Myouren Temple turned out to specialize in martial arts. They even made a game mechanic out of Buddhism being associated with melee attacks. Oddly enough, Taoism is associated with lasers.
Belkar: Hey, listen buddy, sure you can hire Friar Tuck over here—
Monk: Not that kind of monk, actually!
Discussed in Rob and Elliot, when Rob proudly admits he beat up some Buddhist Monks, who weren't as tough as he expected. Elliot informs him that Buddhist Monks are pacifists, and he was thinking of Shaolin Monks. Rob scoffs, then his face falls.
Impure Blood Has Dara, although its unclear if her skills are due to her monk training or not. Her other abilities were present before her training.
Subverted in the Futurama episode "Godfellas": Leela attempts to challenge a group of monks for their radio telescope (so that she and Fry can use it to find Bender), and the monks immediately assume fighting stances. Leela is about to back down when they reveal that they are strictly non-violent and practice martial arts solely as a form of meditation.
In the Middle Ages, retired soldiers would often enter monastic orders, and would earn money for the community giving lessons. Several midaeval and renaissance manuals of arms show monks as teachers. Young nobles, not intended for monastic life, were often sent to monastic communities for all or some of their education, and they'd have to keep up with their combat studies; a community with a resident master-at-arms would attract more pupils.
Truth in Television: Meditation, now mostly attributed to Buddhism and Hinduism, is also a part of some Christian monastic practices (though it's not as prominent as it is in the Eastern religions).
Most prominent would be the Society of Jesus or Jesuits, founded as an order by ex-soldiers. Daily mediation is the foundation of their order with a military discipline.
Not prominent in the sense that we don't usually hear about it, but meditation is a very important part of Monastic life. Everyone who follows the Benedictine Rule must meditate five hours a day; except, this meditation is composed of mostly reading scripture, Gregorian chant, and Psalms rather than the Yoga-lookalike most people think of when they hear the word "meditation".
There were many orders of Warrior Monks during the middle ages, of which The Knights Templar were only the most famous. They used swords and armour, though, rather than fight unarmed—they were both knights and monks, which is a cool resolution of the old question of what to do with younger sons. The Templars were given special permission from the church to use weapons.
Unarmed combat training always went hand-in-hand with weapon training, so a monk knowing one could pretty much be expected to be familiar with the other.
Many Knightly Orders qualify. Notable are the Order of Saint John (also called The Knights Hospitallers). Originally an hospital run by monks, they later became known as Combat Medic and as a full on military organization.
Worth noting that the Order of Saint John still exists today as the Sovereign Military Order Of Malta. They even operate a small number of military aircrafts (most loaned by the Italian Airforce) to carry humanitarian missions. It retains its claims of sovereignty under international law and has been granted permanent observer status at the United Nations.
Even the non-combat orders of monks were populated in large part with younger and bastard sons of minor nobility, most of whom would have received some combat training as part of their education; as well as disabled and retired soldiers.
There was a story floating around a while ago that St. Thomas Aquinas and several other such figures taught their students boxing to prevent them from resorting to knife-fighting. It would imply that these priests knew a bit about boxing...
The ultimate example surely has to be Father Sergio Gutiérrez Benítez, who for 23 years wrestled professionally in Mexico as 'Fray Tormenta' ("Friar Storm"), in order to fund the orphanage he runs. As such, he was the very loose inspiration for Nacho Libre, but that's hardly his fault. Apparently, he still wears his mask for his day to day Priestly work.
Think we also know him as King from the Tekken series. Because when you think 'priest who runs an orphanage', you think 'Tombstone piledriver'.
While they may or may not have had the training to go with it, many early Christian monks were no strangers to hand-to-hand combat: in 6th century Alexandria, if there was a tavern brawl, or a riot, it was even odds as to whether it was started by the "organized" gangs of charioteers, or by monks disagreeing over a fine doctrinal point...
Not that monks as the next millenium knew them had even been invented at that point.
On a related note: Shenoute of Atripe, leader of a large Egyptian monastery in the 4th and 5th centuries CE, was known as a harsh and violent disciplinarian to his charges. He was also not above riding into town on a donkey to beat corrupt landlords and ecclesiastical opponents with his staff. Some of his monks are reputed to have worked as thugs in the employ of the bishop of Alexandria, intimidating his political and theological opponents, and sometimes even attacking them in the desert.
Sufi orders in some districts during periods of political decentralization have had a history of developing relationships with local tough types and/or taking on lots of particularly tough disciples in order to keep order and/or take tithe.
The Safavid kings of Persia, first to unite Iran, descended from a bizarre one of these that developed to slightly cultish levels over a matter of generations before conquering the plateau. The man who did this was fifteen at the time, and also showed the good sense to convert his people to a more standard variant of Shiism than the bizarre messianic one around which his order was actually founded.
The Taliban also originated as a shot at a modern iteration of this tradition in the midst of civil war, although without the Sufi part. As recruitment and mortality increased, the degree to which a given member was actually talib decreased. Although the fact that the early ones actually were refugee taliban and had spent most of their lives without stable home life or much exposure to women in any capacity was almost certainly part of the problem with the insane propriety laws in the areas they pacified.
Of course, there being no particular martial education involved in any of this, this is more Church Militant than kung fu.