I was going to come in and say it would have been called Star Trek, but then I remembered there was a Star Trek episode which was a backdoor pilot for an (unproduced) American Doctor Who ripoff. So the correct answer is if Doctor Who were American, his name would be Gary 7.
http://en.wikipedia.org/wiki/Assignment:_Earth
Unfortunately that's one of the TOS episodes that I've never watched, so I can't comment much on how the character actually played out, but there's your American first Doctor right there.
Edit: It's apparently a better example than I thought it was, since despite the similarities, there's actually very little chance that Roddenberry would have known much if anything about Doctor Who at the time, and what's more, most of the serious similarities didn't actually come into play until the John Pertwee era of Doctor who, which was still in the future when this was made. The BBC probably didn't rip the episode off for that era either, since Doctor Who wasn't the only show in this equation that took a while to get across the pond -- Star Trek took its dear sweet time, too.