“I’m not married to the baby’s father and he is in the navy reserve does he have any rights even if I don’t put he’s last name on the birth certificate, and I don’t want child support or anything from the father?”

If he pursues his legal rights to be a father, than yes. The Courts hardly ever agree with the reasoning that denies a child his right to be supported and to know his family, anyway. If a parent has the desire and ability to parent then the child should receive the benefit of that relationship.