"Hurr durr guns baysplosions eagles everywhere"
Do people learn about America via SpikeTV or something?
Seriously though: They show would be a lot more serious. Americans can have comedy and they can have drama, but both at the same time just doesn't click. Having plunger-robots in a setting as stories like "The Waters of Mars" wouldn't fit with the American idea of genre separation. The doctor would lean much more heavily towards his depressing, angsty side, and the stories would be much more (psudo)realistic. Events would also probably be smaller scale, since Americans like having canon basically reset after every episode. The known existence of aliens in the "modern" time period in the series could never be allowed. Whatever happened, things would be back to normal for 21st/20th century America by the end of each episode.
Contrary to... well just about everyone else in this thread, I think they'd make the doctor full pacifist. People seem to forget that the Doctor is not only quite capable of killing people, but has probably killed more than anyone in the history of time. There are moments when he seems downright cold, cruel even, taking vengeance on those who cross him.
An American audience would probably prefer him to be more Batman-like, with a strict no-kill policy, even when it doesn't make sense. We love our violent heros, sure, but when we have one that isn't violent we like them to be completely that way. The American fandom of Dr. Who likes to idealize him, ignoring his obvious history of genocide in favor of the image of a nice charming British guy who'd never hurt a fly. While the dualistic nature of the doctor is a key part of his character, I think it's one a lot of American fans would rather not have to deal with.