One issue that comes up frequently regarding custody issues from unwed parents - whether a father asking what his rights are, or a mother asking about her responsibilities to the father - is that, though the father hasn't legitimated the child, "his name is on the birth certificate." People often - mistakenly - believe that this means the father is the father for all purposes in the eyes of the law.
I understand this confusion - you've got one of the most official documents in a person's life, issued by the state, naming the father of the child. It seems like this would be the state's recognition that the father is in fact the father. But it's not.
Georgia's laws regarding child custody and child support were written with a general assumption that the parents of the child were married at the time of the child's birth. How we handle cases with unwed parents was later (poorly) tacked on to that framework, which is problematic given that 40.2% of American children are born to unwed parents.
One of the things that hasn't been addressed in this gap is the birth certificate. In order for an unwed father to be named on the birth certificate, he and the mother must give their consent. While there are legal ramifications to this, they tend to be more of the procedural and technical variety, and have little to do with custody and support.
Being named on the birth certificate doesn't grant the father any right of custody, and without more it doesn't create a child support obligation. In order for there to be a right of custody or visitation (again - this only applies to unwed fathers), the father must first legitimate the child. You can read more about legitimation here, or call or email The Lilly Law Firm to get answers regarding your personal situation.
But in the meantime, don't assume that legitimation is a given if the father has been named on the birth certificate, and don't think that rights and responsibilities exist for yourself or your ex that don't.