Legal Question in Family Law in Texas

I think you're asking if he can relinquish his parental rights.

A judge would have to sign off on an order approving such - "The Court finds and orders that the parental rights of John Smith to little Rodney Smith are terminated."

I have seen more than one case where Mom and Dad went to the Judge and said, "We agree that Dad can relinquish his rights, and won't have any more obligation to the child." The Judge said NO.

If your husband only wants to waive visitation rights, that's no problem. He simply doesn't show up for visitation. Court-ordered visitation is always optional; Dad is never ORDERED to come pick up the child. If he does do so, Mom is ORDERED (in the decree) to surrender the child.