And even though he may be able to do all his own stunts, they wouldn't let him; if the main actor gets injured doing a stunt, the movie is held up for months while he recovers.
There are a number of movies where the lead actors do, in fact, do most of their own stunt work. For instance, Tom Cruise genuinely was climbing on the outside of the tallest building in the world in Mission: Impossible: Ghost Protocol
(though he was on wires that were digitally erased in the film). Most of the actors in the Matrix
films did the majority of their own stunts, as did the cast of Crouching Tiger, Hidden Dragon
, as well as Jennifer Garner and Will Yun Lee in Elektra
. Chris Evans did a great deal of his own stunt work in Captain America
, and indeed it's mentioned in the DVD commentary that he had to do all his own running because he has a very distinctive run that nobody else could mimic.
Of course, in most of these cases we're talking about people who were actors first and then trained to do the necessary stunts. Not a lot of people start out as stunt performers and then become actors, though it's been known to happen (e.g. Jock Mahoney, Doug McClure, Ann Robinson, Zoe Bell, and of course the ultimate stuntman, Jackie Chan).