The doctor career has 'deliver a baby' as the task that gets you to the next level in the career. I read that you have to wait until a pregnant sim enters the hospital randomly... but, it's been a very long time, and that hasn't happened. I even played some other families just so there would be pregnant sims in the game, and it didn't help. The task comes up in the list once in a while, but that doesn't help either. Should I give up and pick another career for her?

It's a known bug. I had a sim stuck there for an annoyingly long time before I looked it up. You can use "careers.promote doctor" without quotation marks to get past it. I think you might need to have testingcheats on for that. If you don't like cheating, fulfill all promotion req's, wait until they do another surgery and then pretend that was the delivery and move on.

Eventually a pregnant sim will come along. I've had to wait ages in one game, and then it happened really quick in another family.