As I understand it, it is standard procedure that a doctor of each specialty will be on-call for each night. Whenever a case in that field arises, that doctor needs to come to the hospital. Was there any need for a specialist in this specific case? No, even the DIAC EMT-Bs could have handled it without any real difficulty. I'd wager 99+% of the time, no OB/GYN is needed to deliver. However, there could very well have been a number of emergency situations that arose at any given time that only an OB/GYN is capable of handling, as TheProwler said. So that's why you call the specialist over, just in case. And its not just regarding the delivery itself, but a lot of conditions that may potentially endanger the mother and child after the delivery occurs.
My guess is the doctor gets a page in the middle of the night without much details about the case itself. Its just the rule that when you receive this page, you get to the damn hospital ASAP, and yes of course you respect a cop if he pulls you over (no one is arguing this point). But the point you guys are trying to make about specialists is pretty fucking retarded.