My baby's father said he would give up his rights to her, As we live in the state of Va. Would he still have to pay child support...

First, please pay more attention to posting guidelines. There is a Read First sticky at the top of the Children forum where this was moved from, that directs questions of a legal nature here.

Had you you checked this forum you would have also found a stick that deals with this issue and answers your question.

He will NOT be able to give up his rights. Only a court can terminate rights and courts are very unlikely to do so. And they will not do so to let him out of paying support.