My so-called friend deleted me on Facebook because of an accidental slip-of-the-tongue. I told her I didn't like her cousin. I thought telling the truth even though it hurts was the right thing.
In addition to deleting me on Facebook, she talks about me behind my back and when I try to apologize, she ignores me. Because she talks about me to my other friends, I'm scared I'll lose them too.
I don't quite see this as a "slip of the tongue" on your part. You felt comfortable enough with your friend to tell her your honest feelings about her cousin. If your friend didn't welcome hearing what you had to say, she should have told you so. Ending the friendship was overkill.
Cutting you off the way she did, with one click on Facebook, suggests that she might have been looking for an excuse to sever ties with you. My advice: Forget your so-called friend and don't make any more apologies.
Since you share friends in common, act cordially when you see her, and refrain from making negative comments about her or her cousin. Your mutual friends will be able to judge you for who you are, not who she says you are.