In the state of va. If someone other than the biological father is named, does the biological father have any rights?

Yes, so you and another man committed fraud on the birth certificate ( official government document)
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The bio father can file to challenge paternity, using DNA, he can ask for custody ( claiming a fraud was committed against him) or ask for joint custody. Or at least ask for visitation.

If the court accepts his paternity, he will have full rights as the father

A bio father ALWAYS has rights, unless and until a court terminates them. If the bio father is left off the certificate they have to go to court to enforce those rights.