Making the man responsible is one of the reasons for state sponsored marriage.
Without a marriage license, the man should have no say over and no obligation to any child that reasults.
Or let's do away with state sponsored marriage and use DNA testing instead.
If women have the "right to choose", men should also. Upon formal notification that he's about to become a father, an unmarried man should be able to make a decision as to whether he will support the child, having at least a few weeks to think about it. The woman can make her own decision from there. (Marriage would be considered an explicit "yes" decision)
And no man would be forced to pay support for a child not biologically his, except if he formally accepts the child in full knowledge he's not the biological father