He didn’t say his original transgressions were bad enough to make him a colossal douchebag. He said refusing to apologise for them turned him into one.

While I'm sure not intended, this comment sounds quite sanctimonious.

While I'm not the OP, I would say that the answer to your question is that it doesn't require being intentionally cruel. People in general will (metaphorically) step on each other's toes from time-to-time. If a person is unwilling to apologize over even a small matter, that suddenly becomes pretty rude.

How do you know the other person doesn't see you as cruel, or selfish, arrogant, self-centered jerk, or simply douchebag?

This whole thing is about accepting other possible viewpoints if you don't want to end up as social outcast. I mean we're already in the territory of not-very-balanced people that hurt others enough to have apology expected and don't figure they actually need to apologize on their own (or are too lazy/afraid to do so).

Actually, I take back what I said. I was wrong.

Apologies are useful and not necessarily indicative of cruelty. It acts like a social lubricant and I can understand using it tactically, just to keep the peace.

For some reason, I was only thinking of genuine, remorseful apologies, as if OP was just harassing people all day long, but obviously that's not the case here. There's a lot of small gestures one can do to avoid social faux paus.