Yes, you have to show your license. But no, you don't have to prove you're a citizen to get your license. The recent flap about changing the law to allow illegal aliens to get drivers' licenses has obscured the fact that states (quite rightfully) allow legal aliens to get licenses, and those licenses to my knowledge have no markings on them that denote the holder is not a U.S. citizen.
In CA you do not need to show a DL. Of course illegals can get them. All you need to do here is to say your name. Your name should be on that list. That is it.