My wife and I just had our first child at Phelps Memorial hospital in Sleepy Hollow. We also chose the Pediatrician that's in the same facility.
The Pediatrician never came to the nursery, though he did call to inquire about the baby (he was in special care) and got an update that way. But mfenimore is right - I originally thought that they needed privileges at the hospital where the child is being born, but it's not necessary and they rarely visit the nursery.
As I have found, if there are issues with your child, the nursery will have a Neonatalogist on staff that will attend to your child.
After we were discharged, the next day we went to the Pediatrician for a check-up. It was basically a question & answer forum for me and my wife to ask the doctor, which was very beneficial. The Owners Manual (for the baby) couldn't be found.