https://www.sos.ca.gov/elections/political-parties/no-party-preference
Regarding the CA Primary Election:
As a voter who declined to provide a political party preference, you are considered a "No Party Preference (NPP)" voter, and your ballot will not have presidential candidates on it.
If you want to vote for U.S. President, you must request a ballot with presidential candidates from one of the following parties:
American Independent Party
Democratic Party
Libertarian Party
Um...what about voting for a Republican candidate?
It literally tells you on that page that if you want to vote for a Republican candidate you need to register with the Republican party.
^^^ It's so they know how many votes to make up for, we even have our own color coded election envelopes so the postal service knows which ones to throw out. ;-)