When Jack insists that Tom apologize to Harry.
This trope comes in two forms, based on where he gets the ability to insist:
Jack is The Leader, or parent, or other authority figure and insists on it to smooth over relations. (Justly or unjustly. A Beleaguered Bureaucrat or a Corrupt Corporate Executive frequently insist on it because it seems the quickest way to make a problem go away. Similarly, a parent or teacher who doesn't want to go to the trouble of figuring out who is actually at fault might order an unwarranted apology as well.) Another variation is a Mean Boss forcing an underling to take the blame for the boss' mistakes.
Tom is deeply apologetic to Jack, even though Harry was far more gravely wronged by the matter. Usually Jack refuses to accept his apology until he does it. This is often Jill insisting that Tom apologize to Jack when they are in a Love Triangle.
In Bleach, Ichigo's little sisters demand that he apologize to their cousin Tsukishima after he punches him. The messed up thing? Tsukishima's the villain of the arc. They only think he's their cousin after he messed with their memories.
Wanda (the woman, not the fish) in A Fish Called Wanda demands that Otto apologize to Archie for assaulting him. This is while they're technically on the same side.
In L. M. Montgomery's Anne of Green Gables, town busybody Mrs. Lynde comments on Anne's red hair, which leads to Anne flinging all sorts of insults in her face. Marilla is horrified and orders her to apologize. Anne then presents Mrs. Lynde with one of the most melodramatic apologies ever thought up.
In L. M. Montgomery's The Blue Castle's Back Story Valancy was ordered to apologize once to her cousin for something she didn't do. When Valancy is rude to her during the course of the book, her mother tries again, and Valancy says that that apology would have to do.
In The Vor Game, while regaining control of the Dendarii Free Mercenaries, Miles Vorkosigan meets a demand that Tung apologize to another officer. Miles insists on it, and then insists on doing it for real after a Backhanded Apology.
The Big Bang Theory: Leonard and Penny have an argument, and Sheldon tries to get each of them to apologize to the other because their fight is disrupting his life. He doesn't really care who apologizes to whom, he just wants the fight to stop.
In another episode Penny tells Sheldon to apologize to Leonard for something, but he feels he has nothing to apologize for because he's right. She suggests that he apologize "sarcastically"; since he doesn't do sarcasm well it comes off like a regular apology.
When Penny finally chews Howard out for his lechery, he has an emotional breakdown and locks himself in his room. The A Plot of that episode focuses on the rest of the guys needing to amp up their robot, for which they need Howard, the engineer. Leonard forces Penny to apologize to Howard, though he comes around too late to fix the robot.
Played with in the pilot episode of The West Wing; after making a rude comment to a religious right figurehead about the lack of intelligence of her and her followers on national television, Josh is basically ordered to apologise to her by everyone he comes into contact with that day, and reluctantly agrees to do so despite feeling he was right and justified. After his apology, however, the woman's arrogance starts rubbing Toby up the wrong way until she makes a rather snide comment he interprets to be anti-semitic, at which point he starts getting into it with her, and the meeting quickly breaks down until the President basically shows up rather awesomely, reveals that his granddaughter has been targeted by pro-life extremists, and then tells the religious right representatives in no uncertain terms that there'll be no apology from the White House and, until they denounce the extremists within their ranks, they can pretty much get stuffed.
In the final season Josh (who is heading up the Santos for President campaign) is pushed into a room where Donna (who worked for one of Santos' opponents in the primaries) is waiting.
Lou: I don't know what the problem is between you two, but she's great on television and I don't care if she worked for Francisco Franco in the primary. Right now it's all hands on deck, so work it out.
On Babylon 5, Sheridan is ordered for political reasons to apologize to the Centauri for defending the station against a Centauri attack. He prepares a truly epic Backhanded Apology for them. It's just too bad he never gets to deliver it.
This is the focus of Sports Night's "The Apology". Dan gives a magazine interview in which he says marijuana should be legalized—on the grounds that drugs should be a health care rather than a criminal issue, but this is not how it comes off—and is ordered to apologize on air. The apology he ends up giving is to his brother, who's been dead for eleven years after getting into a car accident while high, due, Dan believes, to his own bad influence.
In House, Cuddy nagged House incessantly to apologize to a patient. House finally "accidentally" stepped on the patient's foot with his cane and acted embarrassed and apologetic about that. Cuddy didn't see the "step on the patient's foot" part and assumed House's "I'm so sorry" was the apology she ordered.
Jedi Academy: Downplayed and then downplayed and inverted near the beginning. Kyle Katarn makes both Jaden Korr and Rosh Penin go through a training course. Rosh figures he can "win" by slowing Jaden down by setting a lightsaber training droid to attack him. Afterwards Kyle admonishes him and doesn't quite order an apology but shames Rosh into giving one. Jaden is disinclined to accept the apology, but Kyle persuades him/her (again, not quite orders) to do so anyway, making it an inversion as well.
In the episode, "Cuckoo for Coco Cards", she demands an apology from Bloo for stealing the attention from her. She gives him a pen, an apology card, and a self-addressed envelope. Bloo refuses to apologize to her and tries to collect all the trading cards of the Imaginary Friends without her help, until he finds out what his status is on his card Big Insensitive Jerkface.
Every parent with at least two kids, and probably every elementary school teacher, ever to exist.
More of a downplayed trope with a job, a boss can just as easily fire you as accept an apology from you; though if you catch them in a merciful mood it's best to act on that mercy and apologize if you want to keep your job. This mindset is especially strict in a military setting as a soldier that misbehaves one too many times will be court marshaled. And God help you if you happen to be in politics and you get caught in a scandal.