Your counterpart wants an apology and you are just dead set against it.
So you can give a multitude of excuses, often many of them childish. You can cite your principles and refuse to budge from your position, Principles are funny things and often are roadblocks to a discussion.
Look, if you really owe an apology, give it. It can go a long way to satisfying the other sides’s feelings. It’s an inexpensive way to get the other side to talk.
Is it about your pride or is it about getting what you want?
“I am sorry that I got so upset.”
“I am sorry that I made you angry.”
“I am sorry that we seem to have a communication problem.”
“I am sorry the we seem to be stuck on an issue that I would have hoped we could agree.”
William Ury: “Don’t get mad. Don’t get even. Get what you want.”