[user104658]

If he's on the birth certificate he'd automatically be "first choice" unless the mother had a good reason for saying he shouldn't be, e.g. homelessness, substance abuse, violent criminal, etc.

If he's NOT on the birth certificate the child would likely go into the system, at which point there could be a DNA test done, after which I'm assuming he would have custody rights (again, unless there is some sort of reason for this not to happen as above).