One of The Punisher Max series Christmas specials had this plot.
In the nuclear holocaust movie Threads, Ruth gives birth in a stable because a guard dog prevented her from reaching the nearby farmhouse. She is then seen in a mock nativity tableau, sitting around a fire with other survivors on Christmas Day.
The Spanish movie The Day of the Beast hinges about The Antichrist being born on Christmas Eve. In Madrid. Under the KIO Towers.
In Rabbit Test, Billy Crystal plays a pregnant man on the lam who eventually gives birth on Christmas Eve while his gypsy girlfriend and a wacko doctor attend him in a schoolbus in a junkyard. The girlfriend's three brothers are dressed in magi robes outside.
Happens at the end of the French comedy Les Rois Mages (The Three Kings).
The 1945 theatrical short "Star in the Night" used this as its main story.
Amazingly enough, averted in the Heliand, an early Saxon adaptation of the Gospels. The author was apparently reluctant to portray Mary and Joseph, both established as nobles, as going without shelter in their hometown, and so omits this detail from his work.
In the episode of EastEnders broadcast on Christmas Day 1990, Disa O'Brien gave birth in an abandoned building.
Scrubs has Turk delivering a baby of a teenage runaway late on Christmas Eve, but who escapes the hospital just before going into labour - Turk finds her by following a star, and after the birth, everyone gathers round for a nice Christmas carol. Awwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww...
Everyone noticed the coincidence and hung a lampshade on it, except for London who didn't get it.
Happens in the "An Angel on the Roof" episode of Touched by an Angel. The episode has several groups of characters being brought together by the angels at a motel on Christmas Eve, including an illegal and pregnant immigrant couple. It climaxes not only with the wife giving birth, but with everyone staging a full "living Nativity" tableau that protects the couple from arrest and deportation (they were related to the motel owner; the angels manipulated circumstances to unite them).
Nightingales has a heavily pregnant woman named Mary arrive on Christmas Eve at the office block where the show is set. After assuring them she's not an allegory, she then proceeds to give birth to, among other things, a toaster, the Pope, Harold Pinter and a tandem.
Walker, Texas Ranger not only has this, the happy couple are named Jose (who gave up a life of crime in a gang and is now running a woodworking shop) and Maria.
That's fairly subtle for this show...
Done in an episode of Moonlighting. It included a trio of FBI agents, all of them named Wiseman.
Newhart had an episode where it was Christmas Eve at the Stratford Inn and a guest ended up giving birth. Luckily for her, there happened to be a group of snowbound physicians staying there as well. In the tag, three brothers named "Weissman" showed up to check in.
Lampshade Hanging: Bob to Mary and "Joe": "Speaking for innkeepers everywhere, I think we owe you one."
Baby William's birth and mysterious conception on The X-Files holds allusions to the Jesus birth story, though it takes place in mid-May rather than December. After going through a pregnancy that shouldn't have physically been able to happen, Scully is whisked away to an abandoned town by fellow agent Monica Reyes to give birth to her son, who is wanted by Super Soldiers intent on kidnapping him. While going out for a cigarette, Monica spies a proverbial "Star of Bethlehem", which Mulder eventually uses to find them. At the end of the episode, the Lone Gunmen step in as the Three Wise Men, visiting Scully and the baby at her apartment bearing gifts. Not only a parallel to Jesus himself: prophecy put him as the saviour of the human race from impending alien invasion. Poor little guy.
Parodied in the Bottom episode "Holy", when Richie finds a baby left on the doorstep of the flat during their Christmas party. True to form, following a few similarities to the Nativity story he lets this go completely to his head. Convinced that he's the "Mother of God" and has maintained his virginity because he's "better than everyone else in the world", he threatens that the other characters will be sent to Hell by his 'husband'. It turns out it's the grandson of their landlord, who left him on the step because he couldn't be bothered dealing with the kid himself.
The season nine NCIS episode "Newborn King" put Gibbs in the role of Delivery Guy to a Marine lieutenant being pursued by mercenaries. And he pulled it off while Ziva fought and killed the last merc in the next room.