I think a basic function of a sincere apology, aside from any spiritual implications, is to communicate to someone whom we have hurt that we are acknowledging having hurt them; which implies that we, having taken cognizance of that fact, intend to avoid repeating the same action(s).
For the very same reason, refusing to apologize leaves the other person free to assume that we simply don't care about the impact of our behavior on them, and wouldn't be bothered by doing whatever we did again at the next similar opportunity.
Very different social messages.
For the very same reason, refusing to apologize leaves the other person free to assume that we simply don't care about the impact of our behavior on them, and wouldn't be bothered by doing whatever we did again at the next similar opportunity.
Very different social messages.