Parents of minor child not married. Can the father, not named on birth cert. just come and take minor child?

If he is not listed on the birth certificate and was not married to you at the time of birth, or married to you now, and has not proven he is the father in court, no,

He can file for paternity, prove he is the father by DNA in court, and file for custody, or joint custody or at least visits.

Why have you not filed for child support also,