The real issue presented is whether or not he can get custody. He can bring an action to determine that he is the father, will have to pay child support, and may seek visitation. Unless it is determined that it will be in the child's best interest to not see the father, then it may be difficult to prevent visitation. You should consider speaking to an attorney about the specifics of your matter.
DISCLAIMER: This answer is provided in response to a "hypothetical" question and provided for general, informational purposes and does not create an attorney/client relationship. The information presented is not legal advice and may change based additional information and research. It is recommended that you speak to an attorney to discuss your specific legal issues.