United States First let's clear the question of legal authority: a father cannot "take" children from their mother. Laws that allowed that degree of authority by a man over a woman have been off the books since at least the nineteenth century.
You have not mentioned
marital status, whether the father's
paternity has been established legally or if there are any existing
court orders. It will be assumed the parents are not married. The presence or absence of a name on a birth certificate is not a deciding factor since paternity can be established by a DNA test irrespective of the information, or lack thereof, on a birth certificate.
An unmarried father has no right to keep the child from her mother unless there is a court order to that effect. He has no personal legal authority to keep the child. In the absence of any order, if he takes the child and fails to return the child to the mother she should call the police.
Generally, if the parents are unmarried the mother has sole custody and control in most states until the father can establish his paternity. Remember, a child's mother can always be identified by medical records. Since the father didn't give birth and he was not legally married at the time of the birth he must establish his paternity by signing the birth certificate at the time of birth (waiving DNA testing rights) which must be done with mother's consent.
If he doesn't sign the birth certificate then he must seek another way to establish his paternity and that is done through a DNA test. A paternity test can be arranged through the court. Once paternity is established in court, the father can request visitations or custody through the court. If the mother retains physical custody she can request that the court issue a child support order. If the father gets physical custody he can request a child support order.