Factual error: When Billy The Kid is taking a bath and dictating a letter to Doc, intended for the governor, he says he is in the Mexican village of Juarez. This town was actually named "Paso del Norte" at the time, it wasn't until 10 years after the events depicted in the film that it was renamed "Ciudad Juarez."
Factual error: The men in the bar are singing "When Irish Eyes Are Smiling" -- which was not written until about 30 years after the events of the film. (00:55:50)
Factual error: When Irish Eyes are Smiling. It wasn't written until the early 1900's by Ernest Ball. He died in Southern California in a theater in Los Angeles in the 1930s, well past the death of Billy the Kid.
Chosen answer: He may have. When Billy started whistling, the bounty hunter's face changes. He continues to speak to Billy in a dismissive tone, but does eventually pull the trigger (although Billy had unloaded the pistol). Hard to believe he would just shoot some mouthy kid in a bar.