Do I automatically have rights to custody on a child I've never met?

Yes, you would have parental rights if you can prove that you are the natural father of the child. If you are on the birth certificate, you should win. Once you have some proof that you are the father, you need to hire a lawyer in the state where he lives and get some advice from that lawyer about how to proceed (I cannot advise you as to any state other than Oregon where I am licensed). However, it would probably be in the child?s best interest to contact the child protective services office in the city where the child lives and get them to do a welfare check. If they see that the child is in a hazardous environment, they may take the child into protective custody. At that point you should be able to file a petition for custody that would allow you to return the child to your home.

You haven't established your right to the child and need to do so in the court in the state in which the child lives. Unless the mother put your name on the birth certificate, you will have to file a parentage action and have a DNA test to prove/disprove if you are the father. After there is clear evidence that you are the father, the court can move toward establishing a parenting plan of some kind. Since you don't know the child at all, it will likely start with supervised visitation until the child gets to know you. If you have concerns re the child's current living arrangements, call CPS in the State in which the child lives.