James Garner did not receive an Oscar nomination for portraying Wyatt Earp in Hour of the Gun.
Wyatt Earp is a historical figure who has been played many times by several different actors usually depicting the events of Earp against the Cowboys lead by Ike Clayton. The role of Wyatt Earp is often portrayed as the hero who wages war against the outlaws along with his more flamboyant friend Doc Holliday (Jason Robards in this version). Earp is technically a slightly thankless role since he is somewhat pigeonholed and even set up in a way to be overshadowed by whoever is playing Holliday. Earp is usually played by the actor known for more steadfast roles which is the case here with James Garner in the lead. Garner is probably best known for the often likable romantic or heroic lead. Garner has just a natural likability in his performances to begin with but Garner actually kinda rejects his usually screen persona here as Wyatt Earp. It would be easy enough to see Garner play Earp with a wink and a welcoming smile but he actually chooses to take a very different approach with the gunfighter.
Garner surprisingly gives a rather cold portrayal of Earp actually and seems to purposefully accentuate the fact that Earp's a killer. This makes sense though for this version of Earp which the film presents as revenge seeking, although still justice seeking as well, since it is pretty earlier on in the film when Virgil Earp is wounded and Morgan Earp is killed. This leaves Wyatt for the rest of the film to avenge his brothers by any means necessary with the help by the seemingly self-hating Doc Holliday. Garner carries himself here with a real intensity here as there is such a lack of warmth in his eyes here which is quite different from the way Garner usually is. It's an effective approach by Garner though as he portrays Wyatt Earp as a man truly hard bitten and changed by what happened to his brothers. He does not have any time to be funny or charming he's on a mission to kill men who wronged him, and Garner bluntly shows this through his performance.
Now the way the story is told in this version isn't really in a way to give a character study while portraying the events. It instead takes a pretty strict stance of meeting each plot point even bothering to go over the courtroom problems faced by Earp and company. The film also spends plenty of time with the other supporting characters although never enough really to realize them all that well leaving the impact Garner can have some what reduced. It's a bit of a shame as what Garner does do in the role is rather striking as he successfully plays this extremely hard bitten version of the character. He also has some nice enough chemistry with Robards, but the film fails to explore the particularly interesting friendship between the two men, something Tombstone handled particularly well. Garner is consistently good here and has one particularly stand out moment where he coldly kills his last man. Garner gives a strong performance and I only wish the film had allowed him to explore Wyatt Earp a little more than it did.