I would vote for socialized car insurance over medicine anyday. Uninsured motorists cause lots of collateral damage and injuries to themselves and others, causing hospitals to absorb the medical costs for unpaid bills.
Instead of destroying an already good health care system, why not just implement socialized auto insurance? We're already forced to deal with the DMV, why not just take it a step further? Socialized auto insurance would also be able to cover those injured in accidents, same as socialized health.
ETA: IF you were forced to choose between health & auto insurance (hypothetical, not literally suggesting it.)