I understand why you're furious, and I, personally, wouldn't apologize to her for anything. What I would do, assuming she was going to be in my life, is just not talk about that specific incident again and be very alert and and at the first sign of her being disrespectful to me in the future, meet it with a "You are not allowed to talk to me/treat me like that. Please leave/I'm leaving." Don't count on her for help or support in anything - just be pleasant as long as she is respecting your boundaries - and those boundaries can certainly include not interfering with your discipline of your children or inserting herself into marital disagreements.
This is wonderful advice. Respectful distance. That's how I deal with a certain in-law. And now she knows not to cross that line.