Is yelling "clear prop" required by FAA regulations?
Like @mongo, I am not aware of a regulation and don’t recall seeing anything about it in the AIM. However, the Airplane Flying Handbook (FAA-H-8083-3B) says this about engine start:


  Prior to engine start, the pilot must ensure that the ramp area
  surrounding the airplane is clear of persons, equipment, and other
  hazards from coming into contact with the airplane or the propeller.
  Also, an awareness of what is behind the airplane prior to engine
  start is standard practice. A propeller or other engine thrust can
  produce substantial velocities, result in damage to property, and
  injure those on the ground. The hazard of debris being blown into
  persons or property must be mitigated by the pilot. At all times
  before engine start, the anti-collision lights should be turned on.
  For night operations, the position (navigation) lights should also be
  on. Finally, just prior to starter engagement, the pilot should always
  call “CLEAR” out of the side window and wait for a response from
  anyone who may be nearby before engaging the starter.