Having been required to navigate several awkward interpersonal encounters in the last few weeks, I think I can distill, into a few words, an appropriate response to such conflicts:
If the disagreement hinges on the other party asking you to compromise your deeply held principles, then never apologize. This is very important.
If the disagreement does not involve any compromise of your deeply held principles, then give in immediately, accept blame, apologize profusely. Send flowers and chocolate.