OK I'm assuming your husband isn't the father? Married or not the biological dad will be allowed to sign the birth certificate. That ring doesn't mean anything if the baby isn't his. He has no rights unless he's the biological daddy.
The baby's father can sign the birth certificate however because you are still married your husband does have legal rights to your baby if he wants to fight for custody he can do that
But I wouldn't worry too much I really doubt he's going to pay to try and steal your child
It differs state to state. I had my 1st child while residing in the state of Mississippi. You can list whoever you like as the father with their consent, however they can only sign their name if your married. So although I had no special circumstance, just being unwed meant he couldn't sign. He is still listed, he may or may not choose to go bk and sign. Not a big deal to us. (Its still good enough for child support btw, it proves he was there and presented ID). So to you I say, Dont complicate the matter by even saying you're legally married. That's not the hospitals business really. He still may not be able to sign anyway. At least he may be able to be listed. My only advice, I'm assuming you are carrying your married surname. Please give the child the last name of his/her father and not your estranged husband. People do that a lot, its weird to me lol... And confusing to the child too!