Can I get divorced in Los Angeles, CA without a Social Security number?

No. There's no requirement for either party's Social Security number on a dissolution filing. You also can't be deported simply because you don't have a Social Security number. If, on the other hand, you don't have a Social Security number, and you're living, and working, in the U.S. illegally, you CAN be deported for that.

This sounds like your ex is trying to gain some kind of a tactical advantage over you. Maybe she is trying to scare you into not responding to her petition so that she can win by default. You do not need a Social Security number anywhere in your divorce papers. You may be asked under some circumstances to provide your Social Security number if you file a fee waiver form, but you do not necessiarily need to provide it.
Good Luck
David C. Beyersdorf

Yes you can get divorced without a Social Security Number.

There is no requirement on any of the paper work for your SSN.