filing a malpractice claim

Is it possible? Yes--a U.S. citizen can sue a resident of or citizen of another nation in U.S. courts if the action giving rise to the cause of action (i.e. the malpractice) occured here; if someone practiced medicine in a given state/county and committed malpractice there, his/her practice and malpractice created a sufficient state interest and sufficient connection to the wrong so as to let the state court hear the matter. 
That said, there are considerable obstacles involved:
1) You have to properly serve someone in another country which, even if you have an address, can be difficult and more costly.
2) Some other countries do not enforce U.S. judgments: even if you sue and win--such as by "default," if the other side doesn't even bother showing up to court--you then can't do anything with the judgment unless that person left significant assets behind in the U.S. Otherwise, if everything he has and all his money is in a country which does not enforce our judgments, you'll never get paid.
3) Even if the other country will enforce a U.S. judgment, to enforce it, you have to work through their courts and their system. At a minimumm, this effectively means additional legal proceedings in the other nation and having to hire a lawyer there; and some countries do NOT have good, efficient, and/or honest legal systems, so you could spend alot of moeny and effort without being successful.
In regards to 2) and 3): which country he/she moved to is critical.
The doctor moved to England--while it would be more costly and expensive than if he/she were still in the States, you could credibly sue.
The doctor moved to India--you are unlikely to ever see any money.