Wife's delivery date immediately after visa expiry

My H1B visa expires on 5th June, 2013… my wife, who is here on H4, is expecting and her delivery due date in 9th June, 2013.

Given that she would not be able to travel back home from US in 8th-9th month, is it possible to extend my stay on medical grounds given the situation?

When you say visa expires, are you referring to visa stamp or petition?

Assuming petition and I-94 is expiring in Jun 2013, you can either file for H-1/4 extension or file for COS to B-1/2 (if the employer is not willing to file H-1 extension or you have maxed out the years).