"To say she should be held accountable for support would mean I acknowledge the relationship, which I don't."
Out-of-wedlock sexual relationships aren't legitimate either, but that doesn't mean both parents don't have a responsibility to the child. I realize that the fleeing lesbian wasn't a biological parent, but she still took on the responsibility and should have to pay something.
Anyone that makes a comitment to care for a child, sister, brother, grandparent whatever has a moral obligation to do so, somehow IMO. Out of wedlock involves a woman and a man. The parents should be the one's legally responsible.