If a woman has an abortion against the will of the father, can the father sue the mother?
The question didn't mention marital status, but since states formally recognize that relationship it's helpful to start there.
tl;dr: The Supreme Court decided state laws that required a woman to notify her spouse were unconstitutional. Thus it's unlikely there'd be grounds for suit.
The central mechanism of Roe v. Wade (U.S. 1973) was a balancing act between what it decided was a 14th amendment right to privacy and the state's interest in both the health of the woman and the potentiality of life. Because Roe explicitly recognized a state interest, Pennsylvania passed a statute in 1982 that required informed consent and a 24-hour waiting period. It also mandated parental consent for minors (with some exceptions) and spousal notification. This reached the Supreme Court in Planned Parenthood of SE Penn. v. Casey (U.S. 1992).
There, the court upheld most of the Pennsylvania law (reinforcing its statement in Roe that a state does have an interest) but struck down the spousal notification portion. To do that, it determined the appropriate test was whether a state was placing an undue burden---a significant obstacle---in the path of a woman seeking an abortion prior to fetal viability. It reasoned that:
- state regulation impacts a female's liberty more than male's during pregnancy (by way of biology)
- if a man and woman disagree, only one can prevail
- not all women are equally impacted by a notification mandate (for reasons of domestic violence, etc.)
Combining this with the notion that women do not lose any constitutionally protected liberty upon marriage, it decided spousal notification would be a significant obstacle and thus an undue burden. In other words, unmarried women don't have spouses to notify, so placing a notification requirement on married women creates an additional burden that the court found undue.
To get back to the question, the father certainly has a right to file a suit against the female (...and it happens from time to time). However, it likely wouldn't go far. Since unmarried women were the baseline in Casey, it's unlikely there'd be grounds for either married or unmarried fathers to sue their female partners. This comes up frequently under the moniker of "Father's Rights," which has gained less traction in the U.S. than in other countries. That said, Wisconsin recently introduced a bill that would allow fathers to proceed against abortion providers.
For there to be a cause of action, the father must have suffered damage. It would be impossible to convince a court that he had.
Further, government (including the judiciary) post Roe v. Wade does not have the authority to interfere in the mother's decision so an injunction is not possible.