State of Florida.
Parents are unmarried from conception through the birth of their child.
Father is not named on the birth certificate.
Couple is no longer together, the father having moved out, leaving the child to live with the mother full time.
Does the mother automatically have full custody by default in this situation? Does the father have any legal custody rights at this point, or will those rights be granted only after he files a case in court?

if the father has not paid child support & abandoned the child to the mothers care. he will have to prove paternity in this case since he is not named as the father. If proven to be the father he will have rights, but not get custody of the child unless he can prove the mother unfit. he can get visitation with the child, but only through the courts. he has no rights other than those that are morally right for the child which should be the first & foremost in the mind of both parents. Having a father in a child’s life is indeed very important ( depending on the character of the father realistically , not because of a mothers anger)*