I hope that's true now. This was back in 2003-2005 and he wasn't asked to sign anything at all. I asked each time if they needed his signature or ID and they said no.
My son (20 months old now) was born in California. A woman came to my room and I filled out all of the forms including my husband's name for 'father.'
He didn't have to sign a thing and he is listed on the birth certificate as the father. I gave my son his last name, because we are married, but even if we weren't - I still could have given my son his father's name.
Fathers are only required to establish paternity when courts get involved for child support & etc.
Oh, I was just talking to my cousin and she reminded me, she and her husband got married after their son was born lol (I can't stand him and blocked it out AND I was in the wedding LMAO). The form is for unmarried couples.