You do not need to have a United States ID or be a resident in order to get married in the State of California.
However you need to have proper Identification. The need for identification has two purposes, to verify that the person is who he/she claims to be, and as an age verification.
Here is a sample list of identification examples that will meet the photo and birthday requirements accepted by a Notary Officiant authorized to issue Wedding License.
State Identification Card
Military Service or Military Dependent Identification Card
Alien Registration Card (Form 1551)
Identification Card Issue by the US Government or Identification Card issued by another country
Matricula Consular Status (must be current and not expired, must have birth date and picture) and a certified and translated (English) birth certificate must accompany the Matricula.
The Marriage department at the County Clerk Office in Los Angeles County may be able to accept other forms of record. Contact them directly.