A lot of people need to be right all the time, even if it is to their own detriment. Therefore they would see apologising as losing.
Social cohesion is quite important and I believe something that is becoming a little undervalued. Especially in a workplace. While of course if is important to stick to your principles, it's not that important to be right in a lot of social situations.
Relationships aren't linear progression. Sometimes you move forward, sometimes you move backwards. Sometimes you can move forward by moving backwards. An apology rarely hurts in the grand scheme of things.
However I would think about what you're saying. And what I'm about to say carries the lack of context from not knowing any more than what you posted.
You weren't listening and it was mostly your fault (your words). She didn't trigger you. And telling her she triggered you will probably not go well. You reacted to her in a way you are now regretting. You need to take responsibility for that and that is likely what you are apologising for.
The conversation about things being fair, save it for later. Put a lot more thought into it.
It's not about being better. It's different.
You've come here asking for advice. Myself and others have given you some. It's up to you to take what you find useful and disregard what you don't. None of us know the full situation.
Let us know what you end up doing and how it goes though, I'm interested and I'm sure at least a few others would be as well.