I think you're unlikely to be fired over this. But you do see the problem you created. You were on sick call. You knew the chance of getting called in was high. You didn't even try to switch it because you thought no one would switch with you. Then you were needed and said you couldn't work. This then leaves the chief in a really tough spot -- they now need to call someone else, and when that person checks the schedule and says "why can't chitown2012 do it?", that's a really tough discussion (if you have a 1st sick, 2nd sick, etc).
That said, your program isn't blameless here. Scheduling women at 37 weeks pregnancy for sick call isn't the smartest move, you could deliver at any moment and need sick coverage yourself.
Go into the meeting understanding that you put the chief in a potentially difficult position, and be apologetic. If possible, it would be great to offer to somehow "make it up" to the other person affected -- but that may be impossible with your training coming to an end but perhaps still an option. It's possible that the other person heard your predicament and was willing to help. It's also possible that their one weekend with their SO is now ruined.