[QUOTE=Just Some Guy]
You’re thinking of Trauma Center which is also available for the Wii, but it’s far from this first doctor game ever released. For example, there were two computer games called “Life & Death” in the late eighties that had you diagnose patients and operate on them.
[/QUOTE]
There’s also Theme Hospital , in which you… design and run a Hospital. It’s a very quirky, British game and you need to have a sense of humour to appreciate it, but it’s also a lot of fun. 
OK, games which will never see the light of day:
Train Robbery: The Age of Steam Travel is at its height, and you’re about to lighten the steam engine’s burden! Do you plant explosives on the tracks to derail the train and loot the wreckage, or are you going to sneak aboard and then rob the guard’s van? How do you get away with your loot? Can you avoid Scotland Yard, the Mounties, or Pinkerton’s? Saddle up and find out!
KAMIKAZE!: Only you can defend your homeland against the Imperialist Roundeye Gaijin! Dive your explosives-laded aircraft onto American warships and strike a blow for the Emperor- but you only get one try, and the entire USN wants to prevent you from completing your mission…
War Correspondent: Relive the greatest battles of WWI, WWII, Korea, and Vietnam from nowhere near the front lines! Sift through piles of censored operational reports and Official News Releases to decide what’s worthy of being telegraphed back to your home newspaper. Minigames include “Getting drunk and passing out with the other correspondents”, “Trying to pick up the hotel bar maid”, and “Not being arrested on suspicion of espionage”.
Watching Paint Dry: The ultimate game of endurance! How long can you stare at the painted surface before determining it’s dry and touching it? To be followed by Watching Paint Dry II: Watching Grass Grow.
Paperwork Filer: Working in an unnamed Government Department, your job is to correctly file each form in its appropriate place! Will you cause a disaster and accidentally file a 417/03B in the file for a 422/07A2, or will you clear your desk by around 4:30ish? You could also try Paperwork Filer 2: Overtime if you manage to complete the original game without losing your sanity…
Mornington Crescent: Can you get to Mornington Crescent, despite the fact it’s a Bank Holiday in Ashford, Wibble has been invoked, the Duke of Wensleyborough rule is in effect, and the little car piece is in Knid on Mayfair with a hotel? 