You might try asking AAA as they seem to have expertise in local laws relating to driving, but I would assume that driving in the USA without a license would invalidate any insurance, including any required state coverage, and could result in jail if caught. A lot might depend on the technicality of whether having lost the license in India means just that you don't have the original or means that you can't legally drive before the document is replaced.

Getting competent legal advice in this situation could be more hassle and cost more than a bunch of taxi fares in the LA area.