Over her own body? Absolutely. Just as the man has control over his. They each have equal authority over their own bodies. What they don't have is authority over each other's body. Nor should they. Your premise of 'control' is that if a man can't control a woman's body, he shouldn't be responsible for any child he fathers.
Um, nope. That's nonsense. As its based on unequal control of one's body and unequal obligation. While our current system is based on equal control of one's body and equal obligation.
In my view the man has to make it clear, prior to the legal end of the time window for an arbitrary abortion if he intends to support the child or not. If not, this gives time for the woman to make a CHOICE, support the kid herself or have an abortion.
And your view is wrong. As his obligation isn't based on his power or his choice.
But on the child's existence. If the child exists, his obligation exists. If he didn't get a say in whether the child was born, his obligation still exists if the child exists. If he didn't have the power to stop the child from being born, the child still exists and his obligation exists.
Power, choice, etc. are spectacularly irrelevant in terms of financial responsibility. And a man's lack of them have no bearing on his responsibility.
The child's existence does. Your argument fails on the basis of control of one's body, unequal obligation, and a false basis of obligation. There's a reason that 50 of 50 States reject your reasoning.