(I guess I omitted the actual answer below. The answer is 'none'. This is about the intimacy presumed of motherhood, which requires a relationship and the interchangeability presumed of fatherhood, which requires a duty.)
This is a continuation of the historical trend of holding men more responsible at a legal level, and women more responsible at a personal level. Abortion is not even the point. Even when abortion was illegal, an unmarried woman who had a child did not necessarily become its legal guardian or mother-in-fact.
Many women's children were raised by their extended families, fostered out to a family friend who already had children, or was put up for adoption. Compensation could be sought by whomever ended up with the child via paternity suit, but only if the father had means.
In many places today a similar option is presented to the mother three times. She can leave the child with an agency through the hospital by arranging this prior to birth. Safe Haven laws intended to reduce child abuse allow her to leave a very young child with a trusted provider anonymously, sometimes up to a year later. And if she considers herself unfit, she can relinquish the child into the foster-care system, which can sometimes then place it with a relative. This is all great -- a self-consciously inept parent is a horrible thing.
But there is no corresponding path for a fiscally inept father. (Most of the men jailed for avoiding child-support are honestly not capable of complying. They may even have income, but they do not have the ability to properly manage it. And they may have gone off and started another family that they can just barely support.)
So this gap in the equality of responsibility is not new. And it remains relevant whether or not you consider abortion an option. If a mother can voluntarily relinquish her parental rights, then so should a father be able to.
She need not have any obligation to this child, but his is established.