If my son's father is in the US illegally and left due to domestic violence but I have not yet filed for custody, does he have a right to see our son?

The immigration status of your son's father is irrelevant.  The question is whether you and your son's father were married at the time your son was born or whether you two have married since your son's birth.  In Georgia, if a child is born out of wedlock, then the mother automatically has sole legal and physical custody of the minor child.  Accordingly, your son's father has absolutely no legal rights to your son (if the child is born out of wedlock) unless he "legitimates" the child in the Superior Court.