I hate the idea that she gets to brush this all under the rug without taking any real responsibility for that.

Tough, man. It happens sometimes. You can't control other people or their actions. The only thing you can control is yourself and how much of an impact you allow her to have on your life. Right now, you're allowing her too much control over you.

And I honestly don't think I'd be able to even feel comfortable around her in social situations without that apology...civility or not.

You're not going to get an apology. You're not going to get a heart-to-heart or a resolution. She's not going to stop talking shit to your mutuals. The best case scenario at this point is that you two decide to be civil to each other in person, but have no other relationship beyond that. I think the action that assures the best case scenario is a short, impersonal note in response to her's.

Your bigger choice, at this juncture, is to either maintain your anger about the whole situation, or decide that it's not worth the emotional energy and literally stop giving a fuck about this person.

Based on your post, your original note, and your responses to everyone, it seems you're leaning toward maintaining your anger. It's ultimately your choice, but I can tell you that is a waste of energy and time. Better thing would be to realize that she's not worth it and, again, let it go.