Not every "problem" can be solved by laws.
Indeed. In this case you've proposed we change a status quo of thousands of years with (I presume) a law taking away a father's rights and responsibilities to his children.
No, with the repeal of laws forcing fathers to act in certain ways. Different cultures already experiment with childrearing duties. The US's method isn't really working out all that well, anyhow: witness 50% divorce rate, questions on punishing child criminals, SS#s required on driver's licenses to track "deadbeat dads", constant abortion debates (with bombings), murders over custody disagreements, paternity suits, talk about getting everyone's DNA, restraining orders, etc.
I am an anarchist so all I'm saying is that it is worth a try to consider children the property of the women that gave birth to them as the best way of dealing with children and their legal implications. These would include crime/punishment, support, abortion and divorce/custody. Up until a certain age or level of development the mother is responsible for the actions of the child. After that, the child is responsible.