In U.S. universities, do male professors get paternity leave?
The answer varies wildly across university to university, even within the same class of university. This can go from the mandatory FMLA minimum of 12 weeks (unpaid) to 12 months, with some pay. I've attached a comparison table from a 2009 survey by Robert Drago and Kelly Davis,  Parental Leave and Modified Duties Policies across the Big Ten. 



I found this from the AAUP's page on family leave, which is fairly useful: https://www.aaup.org/issues/balancing-family-academic-work/leave-policies

In addition to the formal leave offered, there can also be varieties of teaching relief and tenure clock stoppage for both mothers and fathers. All of this seems to vary from place to place, and probably some details could be negotiated at the time of a job offer.