What rights does a father have to an unborn child in the United States?

A spouse has rights, an unmarried parent has rights but they must be asserted, however the man who fathers a child with a married woman (not his spouse) usually can't obtain rights if the couple remain married.

I suggest professional mediation in lieu of or before any court filings.

The rights regarding a pregnancy reside with the mother. The state hasn't yet taken over control of women's bodies completely.

Regarding an unborn child, the father's rights are often secondary to the mother's. Generally, a woman has the right to privacy in her medical decisions. She has the right to decide whether or not to continue a pregnancy. Laws requiring a spouse's consent to an abortion have been found unconstitutional. You need to check the law in your particular jurisdiction.

If you have a particular issue you should consult an attorney in your area who specializes in family law.