If his father isn’t on the birth certificate does he still have rights to my son?
Depends what you mean by "rights".
For example, is there an automatic, unconditional right for the "absentee" parent to have visiting/ access,
that arises out of paying Child Support?
Short answer - "no".
What there is however (as
@Rosscoe alludes to above) is that
the child has
a right to know and have a relationship with their father,
but only so far as that can happen safely.
Yes, the father can apply to have some sort of visiting arrangement with the child.
But such an application is not granted automatically, nor as of right.
Nor is it always and automatically complete and unconditional.
On the up side - you have a good argument that the father is trying to evade (run away from)
his parental responsibilities.
Things like agreeing to surrender his share of parental responsibility,
trying to get out of paying Child Support, but still wanting to intervene, casually, at random,
and only when he feels like it, in the child's life.
What you actually want here is a thing called "
Parenting Orders" from the Family Court.