As the answer by ohwilleke correctly says, a university is not legally required to have or enforce any policy on this subject.
However, a university may, if it chooses, prohibit romantic or sexual involvement between faculty and current students. I believe that some universities and colleges have such policies, and that at one time almost all such institutions did.
I am confident that the is no US federal law that forbids such a policy, and I very much doubt if any US state has a law that forbids such a policy either. The potential for harassment, favoritism, and other abuse is so obvious, that such a policy would be quite reasonable.
In any case, an employer may have almost any policy it wishes on any subject, as long as it does not violate anti-discrimination law, or laws governing conditions of employment, such as the federal Occupational Health and Safety act. If the employee has a contract, as many faculty do, such a provision would need to be incorporated into the contract or at least permitted by the terms of the contract. If the employee is unionized, such a policy would need to be permitted by the tems of the union contract, and would probably be subject to union contract negotiations. But a policy against student/faculty romances is likely to be accepted (although not required) in any of these cases.
I know that many corporate employers have policies prohibiting romantic or sexual relationships between co-workers, and other prohibit such relationships when one person reports to the other.
There is, of course, an inherent power inequality between a student and a faculty member.