When is an apology necessary? I think this is where we move from morals to ethics - with ethics informing the better choice.
If we bring in the question of right or wrong, we forget the purpose of the apology, which is to show both sympathy and empathy. Best case, this undoes the wrong if it leads to a corrected misunderstanding - always a possibility. Worst case, someone apologizes to a selfish person to whom one has committed no real offense and the offense is confirmed in that person's mind.
But then again that was probably going to happen anyway.
The only reason I see not to err on the side of apology is when you feel you're being manipulated more than once into apologizing to a person who is trying to use conflict to raise status.