If the father is divorced from the mother of the child then the father has no rights to the child just because he signed the birth certificate.
Signing the babies birth certificate as the father just proves that the person is the father to the child and doesn't grant them any rights to the child just because the father signs the birth certificate.
To get any rights to the child the father must go into a court and see a judge to get legal rights granted to him.
So for example if the father wants rights to have the child with them at times and see them or do things with them then the father must get a judges court order to grant them the legal right to see the child or have them stay at their home.
The father can still have the child with them or take them to do things alone if the mother allows it but if the mother doesn't allow it then the father has to go to court to get partial custody of the child to grant them rights.