If a 19 year old male and a 15 year old female have a baby, can the father's name be placed on the birth certificate without him going to jail?

The age of consent in the state of Ohio is 16. There is no way for us to know for sure if he would go to jail.

It is highly advisable that he seek professional advice from a local attorney. He may be able to obtain a free consultation from a legal aid office if his income is limited. Any information revealed in a consultation with an attorney would be held in confidence between him and the attorney. Local procedure varies from city to city on these matters. In some states, it is far worse to ommit this information on the birth certificate deliberately to avoid prosecution and/or child support, than it is to step up and take responsibility. Only a local attorney may have insight into local procedures.