"Parents can legally establish the paternity of a child born out of wedlock by
signing a paternity affidavit and filing it with the Iowa Department of Health and Human Services. The paternity affidavit is a free form parents complete and file with the Bureau of Health Statistics (BHS) to legally establish paternity."
Iowa Child Support.
Paternity
How can I establish my
child's legal father?
1. If a mother is married at some time during the pregnancy or at
birth.
2. If a court enters an order naming the father.
3. If both parents fill out and sign the paternity affidavit form.
One of these criteria must be met for your child's father to be on the birth certificate.
If your child's father is listed on the birth certificate, that means you both signed the paternity affidavit, according to what I copied from Iowa DHS. So it sounds like he has parental rights per Iowa's laws and if he wanted to file for custody or visitation he could.
If I was you, I'd be looking up family law attorneys who do free consultations in your area so you know for sure what his rights are. If you're worried about him coming out of nowhere to take him, I'd see about filling for custody so you do have it in writing.