If he is within traveling distance it won't be a big deal. They can transport him to the hospital. I was an XO of a guard unit and had situations like this arise all the time. Just make sure he openly communicates to his chain of command up to the commander. If he is going to be a significant distance away then try requesting some of the options below.
If we were going away I would just transfer their AT to another unit, advance them in an earlier rotation if we had rotations that year or a local duty for 2 weeks.
I recommend getting a note from the Dr. with your due date for his file (not necessary but helps). Also, is this your first child? If so a very small percent is on time. He can also request to do an earlier AT, he could be a part of the advance party so that he could be back closer to your date. What rank is he? Is her new to the unit? Is he in good standing? Has he missed any other duty dates?