No, he has to go where the records are kept with a court order for them to put him on the birth certificate.
My name is Stephen R. Cohen and have practiced since 1974. I practice in Los Angeles and Orange County, CA. These answers do not create an attorney client relationship. My answers may offend I believe in telling the truth, I use common sense as well as the law. Other state's laws may differ.. There are a lot of really good attorneys on this site, I will do limited appearances which are preparation of court documents it is , less expensive. However generally I believe an attorney is better than none, but many will offer a free consultation and a face to face meeting generally will be better, I like my clients to write a short one page history of the fact and questions they have prior to meeting with them, so nothing is forgotten.
When a court establishes paternity, the court order may order that the birth record be updated (see Ohio Revised Code section 3111.13). In my experience, when a court addresses paternity the court does send the order with the form (called an HEA 3029) to change the birth record to the Ohio Department of Health to have the birth record updated. You can also obtain birth records from the Ohio Department of Health or any local health department.
This answer is for general purposes only and does not establish an attorney-client relationship.