Are there any legal requirements concerning airline pilots and their watches? Do they have to be of any particular kind and are pilots required to wear one at all?
Well, a wristwatch can't be used to sneak around the requirement that IFR requires a clock mounted in the aircraft, FAR 91.205(d)(6). A missing or inoperative clock isn't likely in a modern airliner. As far as the FAA is concerned, a wristwatch is nothing more than jewelry.
Ages ago, a wristwatch may have helped with calculating fuel burn and range, but even then, it was a convenience rather than mandated equipment.