It's likely that the watch was simply a practical prop for his character, Cole Younger, a historical figure who was known to be a meticulous and cautious man.
It's possible you might be thinking of a different film, or perhaps you're remembering a scene from "The Train Robbers" where Wayne's character was wearing something else on his wrist, such as a bandanna or a leather strap.
Would you like me to look into any specific scenes or details you remember?