I would like to know if it is possible for a child born in Florida, to have his father's full name (jr) and the father not be on the birth certificate.

You can name your child anything you want but my question would be why would you want to if the father isn't willing to sign the birth certificate?

I should have been more specific. What I want to know is, can you give a child its FATHER'S LAST NAME and the father's name isn't on the birth certificate. The child was born in Florida

You can name your child whatever you choose.

You can use that name, or any name you want,

You can name your child Brad Pitt if you wanted to. But again... it raises the question of why would you want to name your child after someone that doesn't want to be a part of the child's birth/life. It doesn't give any different claim to support or parental rights.