Until there is a court order for child support there is no legal obligation to pay support. You can establish a support order in one of two methods.
You could file a Paternity action claiming he is the Father and requesting support.
The second method would be for you to contact the Department of Child Support Services. They can start a case for you at no cost. It is probably the best method for securing a support order as they can contact the military to enforce the order.
He's not required to pay child support until there's a court order requiring that he pay child support. Whether the baby "has his last name" or not has no effect on whether he's going to be ordered to pay child support.