Can I apply for a commercial-utility licence for an auto-gyro?
Really this is a moot point...

14 CFR 91.319 (a) states:


  (a) No person may operate an aircraft that has an experimental certificate—
  &nbsp;&nbsp;&nbsp;(1) For other than the purpose for which the certificate was issued; or
  &nbsp;&nbsp;&nbsp;(2) Carrying persons or property for compensation or hire.


And again (e) states:


  (e) No person may operate an aircraft that is issued an experimental certificate under§ 21.191(i) of this chapter for compensation or hire, except a person may operate an aircraft issued an experimental certificate under § 21.191(i)(1) for compensation or hire to—
  &nbsp;&nbsp;&nbsp;(1) Tow a glider that is a light-sport aircraft or unpowered ultralight vehicle in accordance with§ 91.309; or 
  &nbsp;&nbsp;&nbsp;(2) Conduct flight training in an aircraft which that person provides prior to January 31, 2010.


21.191 (i) is in reference to operating light-sport aircraft and how it obtains an experimental certificate.

There are very few auto-gyros that are factory built and certified, and none that I can find that are currently being built. 

Anything you build in your home (home-built) needs to pass certain FAA inspections before it is issued an airworthiness certificate and/or experimental certificate. All home-builts are experimental since they are not built under the manufacturers quality control system. Either way, the FAA certifies that you build it to a safe standard and a standard in which the manufacturer intended it to fly. Being used for a commercial purpose, even if that were possible (which it is not) would not change the certification process for that aircraft. It would not require you to abide by any different rules or build standards. That is implying that personal aircraft would be built to a lower safety standard than commercial ones, which isn't the case.