California requires an M2 license to operate a moped. Here's a quick-reference guide from the DMV:
Specifically, (unmodified) gas powered mopeds meet the definition of CVC ss406(a) seen here:
And are subject to the licensing requirements laid out in ss12804.9:
(5) (A) Class M2 includes the following:
(i) A motorized bicycle or moped, or a bicycle with an attached motor, except a motorized bicycle described in subdivision (b) of Section 406.
(ii) A motorized scooter.
(B) Authority to operate vehicles included in class M2 may be granted by endorsement on a class A, B, or C license upon completion of an appropriate examination, except that no endorsement is required for a motorized scooter. Persons holding a class M1 license or endorsement may operate vehicles included in class M2 without further examination.
You can also review the requirements of a class "C" license in that section and see that it does not include mopeds.
Before editing, please link to your sources for why this is not correct.