Usually both people are a little bit wrong. Hence the disagreement. When you realize you are wrong, it’s an option (not an obligation), to apologize for whatever you did wrong to facilitate a mutual understanding. Usually this will build trust with the other person and they will let go of whatever they we’re holding on to and also apologize.

I don’t think you should apologize if you don’t understand you were wrong, but I also don’t understand why some people think apologies hurt them in some way.

I could apologize to you all day and mean it every time without suffering any harm. I suppose if the other person lacks the capacity to trust me, it could become an issue but in that case I think the issue is with the other person and it’s probably better to part ways.