Of course, this answer is just speculation, at least I can't recall any specific evidence in the book to clear this up. My guess is that he is able to use his resume of recent employment, etc., and they don't have to refer back to his degree's date of issue. He could likely just omit that detail and depend on references and his contacts in the current hospital where he works to get him a new job if necessary. When you are that good and apparently well known in your industry, people stop being as concerned about med school as they were when you first got out.
Another possibility may be that he uses his money to purchase new documents proving that he is a doctor. They have a lot of money so they would be able to afford it. They also throw people off their trail by moving around a lot so this helps. They cannot stay in one place for too long because then people will begin to notice that they do not age. Who knows, they may even alter documents themselves. Or maybe they know people who are vampires like them that work for universities and can give him new updated documents. I guess it is all up to our imagination.
If you read Breaking Dawn, they mention that they use J. Jenks to get paperwork illeagaly. Prehaps J. Jenks has made him illegal, but more recent medical school degree papers.
EDIT: I'm sorry I just realized the person above me, posted the same idea, but written much better. But the person they get their illegal documents from is named J. Jenks and they talk about him in breaking dawn.
Their documents could all be fake since they were living in different time periods and they need to shift after a while. They also have a lot of money so it would be easier for them to obtain documents. They are also not involved in a lot of activities, this allows them to not look suspicious.
It's possible that just like all their documents they falsify it by things through people like J. Jenks or they can decided whether or not it's time to change all their identities and go back to school. Especially since they have a lot of money in all different types of accounts with possible different names they can be very influential. It also helps that they that they keep a very low profile as not taking pictures and not being in any involved activities. This keeps them from being questioned.