Hi Jo, I don't know if the Hospital still exists but I don't think they keep birth records. Vital records in California are kept in the County Clerk's office and Health Dept.
The San Mateo Co. Clerk's office can be reached at smcare.org/. I don't know if they will do a search or if you have enough info for them to do a search. Seems like most offices now require the requester to come in and do their own search and I didn't find any volunteer to do this at Random Acts Genealogical Kindness.
I also can't find any information about Ural Morgan. If he moved into the state after 1930, then he might not come up in the on-line databases that I looked in. Do you have any additional information about him?