Since you have only liability coverage, GEICO is correct in saying it's not their problem to collect from your opponent. You need a lawyer's advice, whether or not you wind up hiring him to sue your opponent or his insurance company. (You can talk to his insurance company, but they will certainly try to snow you. A letter from a lawyer will tend to make them behave themselves better.)
If the total damage done was over $500 -- and even a bumper probably costs more than that now -- both you and the other driver are REQUIRED to file written reports with the DMV within 72 hours of the accident. The fact that police wouldn't bother coming out, just means you have to mail it in. If you haven't done it yet, get it done right away or you can be in big trouble, and it might foul up your court case too.
GEICO should get a copy of the report, too, since your opponent will almost certainly try to convince them it was your fault.
California: Land of f***ups 2301Removed crossposted newsgroup...) That's because California's public transit sucks! If California as a whole would actually develop...
Not always; I know someone who was on a jury that ruled the other way. Usually the back driver is at fault, but that rule is common-law (court precedent), not the vehicle code. This is something to ask the lawyer.