Apologizing to the Boss
Some people feel an overwhelming compulsion when they apologize to throw in some kind of qualifying or explanatory statement, such as, “I’m sorry, but I was having a bad day” or “I’m sorry. I wasn’t thinking straight.” Rather than a straight-up “I apologize” they can’t help but follow it up with some kind of exculpatory explanation for their behaviour. Most of the time, this practice is relatively harmless, though of course a person of good character should try to avoid it, but there are some instances where it is absolutely to be avoided — and that is when apologizing to the boss.
I think a lot of people understand this intuitively, but it bears articulation.
The boss does not care about getting into the nitty-gritty details of whatever went wrong. All the boss needs to know is that things are settled now and you respect him. Even in instances where you don’t think you did anything wrong, trying to make him understand “what really happened” may be perceived as a threat to his authority, and a threat to his authority is not just a threat to him but to everyone because he represents everyone and everyone relies on him. In his mind he's probably moved on to the next thing, and forcing your opinion on him is insubordination. Whoever threatens him threatens the group. Every moment the boss is taken away from leading the group is a detriment to the group. The boss quite simply does not have the time to work out the minutia of every single personal interaction he has with his subordinates.
All the boss needs is an apology of few words. This reaffirms the hierarchical nature of your relationship. Anything more may threaten it. The boss is not your buddy. A buddy cares about your point of view, but the boss has no such obligation.
If the boss doesn’t throw you out, that means he still wants you around and thinks you can contribute to the group. I guarantee he will not be wasting any of his time internally agonizing over a disagreement he had with one of his subordinates. He’s got more important things to do. If you show him respect and obey his instructions, that is all that matters.