Yes, he will get parenting time if the court orders child support. Visitation is an old term? :) The idea is that he is to have responsibility for his child. His child is more than a visitor.
If he denies your child it sounds like he wants to believe that he's not the father. In that case wouldn't he want to sign off on his parental rights? Or perhaps he feels that if he signed the paper he would be admitting he's the father and he wants to continue believing he's not?
I'd get the paper that he needs to sign and directly ask him to sign it. Talk with an attorney, perhaps, and get suggestions for ways to explain the signature in such a way that he's able to feel that signing this is just making him not responsible for the child. If his name is on the birth certificate it doesn't matter if he's the actual biological father. He's the legal father and if he wants to say the baby isn't his signing the paper makes that legally true.
If his name is not on the birth certificate then I believe that your husband can adopt him without the biological father relinquishing his rights. You would have to put an ad in the paper seeking the father but that would be a formality if the bio father denies he's the father. You do need to ask an attorney about that, tho.
Along this same line of thinking, if his name is not on the birth certificate, you could possible let him off the hook by saying something like OK, you deny that you're the father, it's OK with me. At a later time you can say you don't know who the father is for legal purposes. If your husband then wants to adopt him you'd have to put an ad in the paper for a specified number of weeks and when there's no response your son is available for adoption by your husband.
You do need to discuss this with a family law attorney. Many attorneys will talk with you for 30 mins. for no charge so that you and the attorney can decide if he can help you with the process. The attorney may tell you what you need to do and then you can do it or he may be able to file with the court at a price that you'd be willing to pay.
LATER: I'm not suggesting lying about not knowing who the father is. I'm suggesting that you can agree with the man who says he's not the father that he's not the father and that would mean "not knowing" who the father is. If he's "not the father" you really do not know in legal terms who the father is. A technicality to be sure but one used in the legal system.
I'm suggesting the same thing that you are Luci. Just using too many words. :)