First I need to know if you are married. If you are he can file for custody in Georgia. At this point, I doubt he could get anything but visitation up here until the child is healthy and a bit older.
If you are not married, he would need to file for legitimation, otherwise he has no rights.
You, however, to have a right to child support under any of these circumstances.
I would be glad to help you and my rates are quite low. 610-999-4502
First of all you haven't said if the two of you are married or not, which makes a huge difference in the answer.
Second, you didn't say if he simply says he wants custody or has actually filed something.
Thus any answer you get here will be incomplete. If he does file anything, get a lawyer IMMEDIATELY. That will be the single most important step you can take and it matters.
If he's not married to you, he has NO rights to the child unless he gets a court to legitimize the child and also gets a court to award him parenting time. If he is married to you he has rights even now.
Bear in mind that the issues will be far more than what happens in the hospital. Decisions for the next 18 years may be made, including support, custody and visitation.
If you find this answer helpful, please mark it here on AVVO as helpful. In answering you, I am attempting to communicate general legal information and am not representing you (and am not your lawyer). Do feel free to call me at 404-768-3509 if you wish to discuss actual representation (the phone call also does not retain counsel; that requires an office visit and appropriate paperwork). In that a forum such as this provides me with limited details and doesn't allow me to review details and documents, it is possible that answers here, while meant to be helpful, may in some cases not be complete or accurate, and I highly recommend that you retain legal counsel rather than rely on the answers here. (You can also email my office at firstname.lastname@example.org . An email also does not retain my office, but can help you get an appointment set if you prefer not to call). I am happy to discuss possible representation with you. Any information in this communication is for discussion purposes only, and is not offered as legal advice. There is no right to rely on the information contained in this communication and no attorney-client relationship is formed. Nothing in my answer should be considered as tax-advice. To ensure compliance with IRS Circular 230, any U.S. federal tax advice provided in this communication is not intended or written to be used, and it cannot be used by the recipient or any other taxpayer (i) for the purpose of avoiding tax penalties that may be imposed on the recipient or any other taxpayer, or (ii) in promoting, marketing or recommending to another party a partnership or other entity, investment plan, arrangement or other transaction addressed herein. I am also required to advise you, if your question concerns bankruptcy, that the U.S. Congress has designated Ashman Law Office as a debt relief agency that can help people file bankruptcy.