We'll never know if prices would be exorbitant under a free market in the US because we've got a nasty mix of corrupt insurance companies, healthcare administrators, and political lobbying (bribery) keeping it from being a free market. Generally we can't even ask the price of a procedure with which to compare providers, and for most insurance, you can't even choose your provider anyways. Does that sound like a free market to you?!?

As the other poster pointed out: Your logic is flawed because your premise is wrong. If the US had state controlled medical, I have no doubt our politicians would figure out some way to ruin that too.