There is a California birth index available on Ancestry.com and familysearch.org. I presume that index would include the birth record under the adopted name.
Ancestry.com allows you to put in the day and month, unlike familysearch which just has a spot for the year, so I would use Ancestry to search the birth index if you can. (If you're in the US, your local or state library may provide free access to Ancestry.)
FYI-
You don't need to be a paying member at Ancestry or go to a local libarary for access to Ancestry to be able to search th CA birth index by Day, Month, Year.
It can be done for FREE at the below site.
http://cabi.mirrorz.com/
