he has no choice in the matter at this time. Once the child is born he can sue to establish paternity (and along with that make him subject to paying child support should the mother care to but he also could get visitation, or possibly even custody) he can ask the court to amend the name. It is likely the courts would, at best, hyphenate the mother's and his name if the mother objected.
Now, if you are willing, you can deny paternity if the bio father will accept paternity before the mother leaves the hospital. That can be taken care of with an Acknowledgment of Paternity.
btw: you don't have to give the child your name (or the mother's name if different) if you choose.

