The general rule is that whether or not the parents of a child are married, both have equal rights to and may file for and obtain, custody, visitation, etc. In addition, depending on the circumstances, either can also be required to pay for child support.
The law is a bit confusing on what the rights are prior to birth and do vary by state. Some courts have held fathers have certain rights prior to birth, but HIPAA and civil rights may not go so far as to allow a father to force a woman to have him in the room during exams, is questionable.
Where a situation between people who are having a child may be stressful, even prior to the birth, it may be best for parties to speak to attorneys in their state, particulary those who practice in the area of "family law" to be sure their rights are as protected as possible.
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