Congratulations on the little boy! That’s what I’m having too. As others have mentioned, usually after natural birth, you leave in about 24–48 hours. That is if everything with you and baby is going well. If there are any complications, it could be extended. If you are okay, but baby isn’t, he would stay and you could be discharged. If he is okay and you aren’t, they would keep both of you admitted (in most cases) so that he would be with you. C-sections generally keep you in for a few more days to be sure that your incision is okay.
Hospital policies vary from hospital to hospital, so you would need to ask them what their policy is. If you haven’t toured your hospitals L&D unit yet, try to schedule a tour and you can ask them their policies then. If you don’t want to do a tour, just call and ask them their policies.
The hospital we will be having our son at not only allows dad to stay, they have a pull out sleeper love seat in every room and all the rooms are private rooms. Also, siblings are even allowed to come (though I don’t think I want my 9-year-old there in the room for the actual birth).