In Sonic X, Amy asks if Knuckles was going to apologize to Sonic for fighting against him because Dr. Eggman had him tricked, only for Knuckles to tell her to zip it just before smiling at Sonic and the gang, tells them "later", and walks away. In season 3, however, he did apologize to the gang for falling for Eggman's tricks.
The Jokerhates apologies. He hates the very idea that something should ever be apologized for.
Joker: "My friend... Jonny Jonny... what I hate more than everything... is apologies."
In The Sandman, Morpheus is the immortal ruler of dreams, and tends to be very inflexible, refusing to apologize or admit error. In his backstory, he sentenced a woman to hell for eternity simply for refusing his advances (when she had very good, selfless reasons for doing so.) After several thousand years, he is finally goaded into freeing her by his sister Death, which sets in motion the events leading to his own demise.
Black Swan, the enigmatic voice of knowledge in New Avengers, is first seen triggering a bomb to blow up an alternate Earth. She never apologizes for this or anything else she says she has done, saying she did what was necessary. But she isn't heartless; the emotional toll her career has taken on her manifests in Bad Dreams.
This has become part of John Stewart's character, almost to the point of repetitiveness. John Stewart killed a planet... and he would do it again! John Stewart killed a Green Lantern... and he would do it again!John Stewart killed a Green Lantern planet... and he would do it again! Note that these decisions are usually the last available option.
In one scene of Wedding Crashers: Jeremy had made a foul remark about one of the proper ladies in the wedding, earning him the glares from the guests. He then claims this statement to John:
Jeremy: I'm sorry. I'm not sorry. Okay? I'm not gonna apologize, I'm a cocksman! *gets glares from wedding guests again* Tourette's.
Granny Weatherwax provides the page quote in Carpe Jugulum, but she embodies this trope whenever she appears. Granny never apologizes, to anyone, for anything, because she's not sorry.
Susan of Changes is very clear and unapologetic about hiding Harry's daughter from him for years. She considered it the best solution then and won't waste time worrying about "what ifs?" Harry understands and tells her simply they are through which Susan expected years ago when she hid Maggie away and isn't sorry for that either.
Satan from Paradise Lost refuses to repent to God due to his pride that had him thrown out of heaven, even if God is willing to welcome him back if he does so, because, in his words, he would rather rule in hell than serve in heaven.
Live Action TV
Insufferable Genius Sheldon from The Big Bang Theory doesn't apologize for anything he does. For him to apologize would imply that he was actually wrong, which is impossible for a genius like him, as his calculations for all things in life are always correct. He could never accept the possibility that he might be wrong, and thus he never apologizes. Of course, when push comes to shove, even he'll break down and try to make amends if he wronged any of his friends. He'll come up with a solution or compromise to the problem that he started, thus still making him look good, but that's as close as he gets to actually apologizing.
In Coach Trip, Bruce and John got kicked out of the fourth season after Bruce picking a fight on the coach for no apparent reason. He then refused to apologize for doing so.
In one episode of Friends, Joey's at dinner with a woman who specifically tells him not to touch her dessert while she's in the bathroom. When she comes out, he finishes the last bite with a smile and says "I'm not even sorry."
In the Home Improvement episode "Luck Be a Taylor Tonight", Fred states that he never apologized to any woman, and when Tim asks how many times has he been married, his response was three because he can't find a woman who understands him.
In House of Anubis, when Joy tries to talk to Nina about the former's kiss with Fabian, Nina asks if it's supposed to be her way of saying sorry. Joy responds by saying, "I wasn't going to apologize,"
Patricia has a moment like this too where she refuses to apologize to Eddie for accidentally spilling a secret of his, only it's justified, because she didn't actually do anything wrong, and was just being forced to apologize to Eddie so she could convince him to drop an article he was writing about the local ghost.
Gibbs from NCIS. His catchphrase: "Never apologize. It's a sign of weakness."
Angela from The Office (US) refuses to apologize for her actions. This is exemplified in an episode in which she is given the task of calling up clients who were upset due to a company mistake. She refuses to say the company is sorry to everyone she calls, even though she has nothing personal at stake.
This is discussed in one episode of The Sopranos after Christopher gets out of drug rehab:
Tony Soprano: So, what step are you at now? Christopher: I did all the steps, except for the one where I'm supposed to go around and apologize to all the people I fucked over when I was using. Tony Soprano: ...I think maybe you shouldn't do that one. You know, let sleeping dogs lie. Christopher: Yeah, that's what I was thinking.
Downplayed on Babylon 5 with the character of Londo Mollari. He's shown to apologize several times throughout the series (notably in "The Gathering" and "The Geometry of Shadows"), but in "The Very Long Night Of Londo Mollari", it's revealed he was never sorry that he'd done something, but that he'd been caught. It's not until he's willing—and able—to give a sincere, unlooked-for apology to his rival G'Kar that he's able to escape the coma his conscience had put him in.
In a stretcher match against Shawn Michaels in One Night Stand 2008, Batista mouths off "I'm not sorry. I don't love you" to him before planting a Batista Bomb. This was an Ironic Echo to Shawn's WrestleMania XXIV match against Ric Flair in where he says "I'm sorry. I love you" before giving Flair a superkick thus not only beating him, but also ending his career. This was the main reason why Batista had started a feud against Shawn in the first place.
Vico from A Dance with Rogues will never apologize for anything he does or has done to the Princess and to others. The only apology he'll ever give the Princess is for raping her, but he only gives it if she slept with him in the Dwarven Inn and then goaded him into finally admitting he's sorry. He will not apologize if not asked right then and will never bring up the subject himself afterwards.
Othar's twitted side-story in Girl Genius has the Master of Paris read Othar a letter supposedly from him, in which he accuses himself of recent crimes in an apologetic tone. Othar's answer?
Othar: That letter is a fraud! Othar Tryggvassen may do things that lesser men find objectionable or slightly illegal, but I never apologize!
In Hellsing Ultimate Abridged, Alucard has caused so many horrible acts such as massive collateral damage, multiple noise complaints, a dozen innocent deaths, and lots of sexual harassment, and after all of this, he states that he's not apologizing for them.
In the episode, "Cuckoo for Coco Cards", Bloo doesn't apologize to Coco for trying to steal all the attention from her, despite the obvious hints that he should, such as a pen, an apology card, and a self-addressed envelope, and thus tries to collect all the trading cards of the Imaginary Friends without her help. It's only after he finds out what his status is on his card Big Insensitive Jerkface that he apologizes to her.
In the special, "Destination Imagination", when World apologizes to Frankie for shrinking her friends, Bloo mentions to him that when Frankie makes him apologize, she makes him sound like he means it (a fact later proven in the episode).
Futurama has the recurring character Hedonismbot - a robot who...was apparently built for no other reason than to be a hedonist. Indeed his first ever spoken line was (in the context of all robots being responsible for Earth's eventual destruction);
In the episode, "Paste Makes Waste", Buttercup teases Elmer Sglue for eating paste. When Elmer turns into a giant radioactive monster as a result of eating a fly that swam in toxic waste and landed in his paste, Blossom and Bubbles get trapped in him, leaving Buttercup to apologize to him the only option of saving them. Of course, Buttercup is reluctant to do so.
In the episode, "All Chalked Up", Buttercup destroys Bubbles' chalk, leading Bubbles to get a new set from a butterfly that is really a disguised Him. The chalk transforms her drawings into giant monsters. At the end of the episode, Buttercup reluctantly tries to apologize, but Bubbles accepts her apology before she can give it.
In the episode, "Fools in April", Squidward pulls an April Fool's prank on SpongeBob and everyone is mad at him for making SpongeBob cry. Squidward does feel guilty, but has a hard time apologizing to SpongeBob, since it's not something he normally does. After several unsuccessful attempts to apologize, he puts his head in a bubble, and as a result, SpongeBob can't hear him when he says he's sorry. He does give SpongeBob a full apology when he remembers what everyone said to him earlier.
In the episode, "Little Yellow Book", The townspeople of Bikini Bottom were mad at Squidward for reading SpongeBob's diary, making him cry. Mr. Krabs orders Squidward to apologize to him, only for Squidward to reply that he'll get over it. Squidward spends the remainder of the episode feeling proud of what he did until his house gets repossessed and he ends up stuck in the pillory in the middle of the town while people are throwing tomatoes at him. He desperately asks SpongeBob to forgive him, until he reads Spongebob's real diary, feeling that it's Totally Worth It while tomatoes are being thrown at him again.
In an episode of Spliced, Entree accuses all of his friends of stealing his lucky stone, only to find out that it was stuck in his ear the whole time. Entree's friends then tell him that he should apologize to them, but Entree refuses because he believes that "only losers apologize."
In The Penguins of Madagascar episode "Out of the Groove," Julian becomes jealous when all the zoo patrons pay more attention to a bunch of dancing baboons instead of him, so he gets back at the baboons by dumping a crate of skunks onto them. The baboons then get revenge on Julian by casting a spell on him that causes him to lose his ability to dance. The baboons say that they'll only reverse the spell if Julian apologizes to them, but Julian refuses because he thinks that apologizing is for the "weak and wrong."
Harry and His Bucket Full of Dinosaurs had the episode "Yo-hoo-hoo," where Harry and his dinosaurs play "Pirates," but Harry soon gets carried away with the game and starts ordering everyone around and being mean to them. The dinosaurs then stick Harry on a deserted island and won't let him come back until he apologizes to them. But Harry refuses because "pirates never say they're sorry."
Leon Czolgosz, an anarchist who assassinated U.S. President William McKinley in 1901 pleaded guilty on his trial and refused to speak to his lawyers. Before his execution, he stated: "I killed the President because he was the enemy of the good people – the good working people. I am not sorry for my crime."