Most Helpful Guy
What you have to ask is whether you value your relationship more, or your own ego.
I am not talking about this one incident. I am talking about the long term prospect of your relationship.
Couples where one person is too stubborn to apologize, often don't work out. Over time, the frustration from one person because the other never admits they're wrong, or apologizes, will make the relationship nosedive.
It's quite easy for most people to be stubborn and refuse to apologize. It take a person of immense character to realize they were wrong, and actually admit it in order to heal the relationship.