One thing I learned early on is when arguing with a non believer, you are wasting your time until they are saved. The old verse about casting pearls before swine comes to mind. They cannot appreciate real wisdom from God if they don't believe God. I've seen people that sit in pews every week that will argue that homosexuality is fine to God and babies aren't people till mom says she wants them.
People that stupid cannot learn anything from Scripture. If she denies what you have said, then she probably has a problem with Scripture in the first place. Many Christians are frustrated when we drop the Wisdom of the ages on someone that denies the Wisdom of the ages. It's silly to waste time. There is nothing wrong with disagreeing in public, but you will NEVER change anything by arguing with them until they accept Jesus as their Savior. Arguing with these people is like pig wrestling. You get muddy and it frustrates the pig. Ask what her relationship is with Jesus and see what she says. That's where I would start.
I liken this incident with her to when my kids were a lot smaller and younger. When they couldn't get something a certain way they would just throw the biggest fit you'd ever seen. After they calmed down and realized that I was trying to protect them, they realized they were wrong and everything was hunky dory.
Granted, this will take a lot longer than a few hours to get through, but she was presented with the Gospel (through other means and not shown in this posting) and now she has a decision to make. She knows that, and only time will tell what path she takes. I know this, she still has a tender heart despite the liberal bravado she puts up. God can reach her. I also reassured her that I would always be there for her, that I would emulate, as well as I can, the love the Father has shown tor me and would love her unconditionally. I think she may have shed a tear or two.
Thanks for your advice. I will definitely take it to heart.