I’m not a person who likes admitting error. If I can avoid acknowledging my mistakes, and just move forward, that’s my preferred method of resolution.
That method is unfortunately not always possible. Some people need the validation that you were incorrect to move forward. You need to grin and bear it.
Here’s what I do to make the crow a little less feathery. I start by plucking off the unnecessary details, that way it doesn’t become an indictment on unrelated topics.
Next, I broach the topic directly.
I’ve thought about it, and I was wrong when I said/did something. I understand I was incorrect because XYZ. Going forward, I hope you trust that I’ll not make the same mistake again.
This acknowledges culpability, shows that you understand the other person’s perspective, asks to move forward, and puts you on strong footing to continue your relationship with the person.