If a unmarried father signs child's birth certificate can they obtain custody of child?

An unmarried father can obtain custody ( depending on situation) even if they did not sign birth certificate.

To gain full custody, most likely they would have to prove the mother unfit, but they are very likely to get Joint Custody unles there is a reason they should not have it

Whether the parents are unmarried or not has almost no bearing on custody. What determines custody is the whether the parent is considered the legal parent. If the father signed the birth certificate or an acknowledgment of paternity then he would be considered the legal father and, by default have joint legal custody.

Whether that father could get physical custody is dependent on a large variety of factors. If you need more help, use the Answer options to add more details about what is going on.

if a unmarried father signs childs birth certificate can they obtain custody of child?

Who is "they"?

A father, unmarried or not, when used as an object of a sentence, would be a "he".