Is health insurance responsible for the ridiculous cost of health care in the US?

Since capitolism is based on the concept that competition mixed with supply and demand will motivate compainies to maximize their efficiency and innovation, would it not seem that health insurance, whereby the person receiving the service is not actually required to pay the bill, would prevent hospitals from caring about havign a reasonable price fortheir service, or for operating efficiently? (sorry for the long sentence) Since hospitals are plentiful, would not simple free market economy work with with medicine in the US, and force prices to drop the a reasonable amount that people can actually afford, as it does with food, shelter, and all the other necesities that benefit from the superior free market economy? Could solving the problems of health care in the US be as simple as eliminating all health insurance companies?