If an illegal immigrant father has a child in the US can he stay the mother is a U.S. citizen?

No. He will have to return to his country of origin and apply to be an US citizen like everyone else. The child is an US citizen because he/she was born to a parent who is an US citizen and also because the child was born in the US. Having a child with a US citizen or while in the United States does not give you US citizenship. My advice would be, first, to return to your country of origin because you are here illegally, then marry the mother of your child and then apply for permanent residency.