My question involves paternity law for the State of: Virginia

My daughter gave birth this weekend to a baby boy prematurely. She and the father were "friends with benefits" when she got pregnant and have had words, had a falling out, whatever you want to call it, and he is refusing to sign the birth certificate and acknowledge paternity, demanding a paternity test. Okay, that's fine; that's his right. BUT at the same time, he is trying to tell my daughter what to name the baby and is traipsing around the hospital and the NICU (the baby is a little bit early and needs to grow) telling everyone he is the baby's father.

He can't have it both ways to my way of thinking. He either is or is not the father. Can my daughter tell the hospital staff that she doesn't want him there and keep him away from her and the baby? And if she does that, if he goes to court later seeking custody (obviously after he has the paternity test and proves he's the father) would the judge look at it as her trying to keep him away from his child? Am I correct in my thinking that he has no legal rights until paternity is proven, either by him signing acknowledgement or proof by paternity test?