That can of worm is already open. Already, courts can comple a man to support a child to term, and in many cases when he can prove it isn't his.
I'm pointing out that there is a downside to this legal debate.
You're also presuming that all women get custody and all men pay child support. That's simply not true. Not only are men getting custody--and not just shared custody, but sole physical custody, but women most certainly pay child support.