Is there any regulation that authorizes autopilot use when cleared for a visual approach?
The general answer is no. A visual approach isn't considered an instrument approach according to ATC so there's no restriction - or direction, if you prefer - on how to fly one:


  A visual approach is an ATC authorization for an aircraft on an IFR
  flight plan to proceed visually to the airport of intended landing; it
  is not an instrument approach procedure. Also, there is no missed
  approach segment. An aircraft unable to complete a visual approach
  must be handled as any go-around and appropriate separation must be
  provided.


And from the AIM 5-5-11:


  The pilot must, at all times, have either the airport or the preceding
  aircraft in sight. After being cleared for a visual approach, proceed
  to the airport in a normal manner or follow the preceding aircraft.
  Remain clear of clouds while conducting a visual approach.


That means that when you're cleared for a visual approach it's entirely up to you how you approach and land, providing that you follow the rules mentioned in the AIM and  - in the case of airline operations - whatever procedures are detailed in your operations specification.

But if you're asking about autoland specifically (not general autopilot use) then the FAA does have a few relevant comments in their guidelines for CAT III operations:


  Autoland or HUD to touchdown operations are required for all CAT III
  operations, and many operators use autoland or HUD for CAT II, CAT I,
  and visual flight rules (VFR) operations as well. Part 121, §
  121.579(c); part 125, § 125.329(d); and part 135, § 135.93(d) prohibit the use of autoland or HUD to touchdown in any operation unless the
  operator is specifically authorized via OpSpecs.


And:


  Practice autoland or HUD to touchdown operations may be conducted at
  CAT I runways in CAT I or better weather conditions where the facility
  classification is unknown, because the flightcrew is monitoring system
  performance, visually verifying the position of the aircraft, and can
  determine whether to continue to a landing or execute a missed
  approach in CAT I or better weather conditions. Though it is not
  recommended, flightcrew monitoring and increased visibility also
  permits practice autoland or HUD to touchdown operations using an ILS
  classified as I/A, I/B, I/C, or I/T, provided the operation is
  performed in VFR conditions


I understand that to mean that autolanding is allowed in visual conditions (provided it's authorized via OpSpecs) for training/practice purposes, and since they mention autolanding use even under VFR I don't see why it wouldn't be allowed on a visual approach.