Is their a legal law in Texas that states the biological fathers newborn child can't be released to him by the hospital ? (unmarried if the newborn can't be released to the biological mother).

Well all laws are legal. Why can't the newborn be released to the mother, that may help in knowing what is going on.

If the father is not listed on the birth certificate he is not the legal father yet. If he is on the birth certificate he is the father and there should be no reason he can not get the child.

If he is not on the birth certificate he will have to file in court for at least temp custody of the child.

1) why can't the baby be released to the mom ?
2) Did the fathers name make it to the birth certificate ?
3) is the mother breastfeeding ?

In the case of umarried parents, custody is generally vested n the mother unless a court orders otherwise.

So we really need more info about the situation to advise.