There is the basic principle that anyone can be sued for the consequences of their actions. Uber arranged that drive. They didn't check for insurance status. That could be cause enough. Plus between you and the driver there exists no business relationship. So you can argue that there's nothing to sue about.
Besides who would you sue ? "The driver" doesn't work too wel. You'd sue Uber to get the name of the driver in the first place.
This go-between-suit is pretty common in legal cases. What should happen is this. Judge agrees Uber owes your medical bills/restitution/whatever applies. Judge also agrees that Uber can get this amount from their driver. Then Uber has to cough up the money, but if they ask for delay in payment until their suit of the driver is finished, 99% certain they'll get that. Then you get the choice of getting the money from Uber, or joining Uber's suit against the driver (not a hard choice with Uber).
