Does a 4-year-old child's biological father have any rights if he is not on her birth certificate or any document and has never seen her?

No, the father, if not on birth certificate does not have any rights what so ever, unless he takes you to court for a DNA test, and proves he is the father then he does not have any rights. If the space on the birth certificate is left blank then what you could do if there is another man willing to take over the right to be the father, you can go back and name the father on the birth certificate, and he will be named the father, and will hold all responsibilies!

Yes. The biologial father has all his parental rights until he is deprived of those right by judicial or legal process. The above answer confuses EXERCISING parental right and HAVING them.

Of course, the question is not specific enough to make the answer accurate in all situations. For example, if the the biological father impregnated the woman while she was married to another man, and her husband raised the child (knowingly or unknowingly) as his own for the period of time set by state law, a legal presumption would arise that the husband is the father that could not be challenged by any person (even with biological proof that someone else is the father). Thus, under those facts the biological father would have no rights.