That's what competition in a free market is, no?
For most of the market, price competition does not occur because providers will not disclose the price of a service before selling it.
There is also an incredible amount of corruption surrounding drug prescription and diagnosis to get people buying way more drugs than they need to.
It isn't anything close to a free market. I could never open a health clinic out of my garage and put out a sign saying free anything related to medicine. By comparison, India has much more volatile healthcare (because they can't police everyone trying to play doctor) but get innovation as a result.
You got hit riding your bike and are bleeding profusely from your abdomen. The ambulance driver asks, "There's a hospital 6 minutes away but they don't post their prices publicly (who does?) and we've never brought anyone of your condition there before. We think it will probably cost between $500-$10,000. But there's a hospital 45 minutes away that we know for sure will cost $400-$800. What do you want us to do?"
Let's suppose the government mandates that all hospitals must post their prices publicly. But you're bleeding to death. Can you really go shopping now?
It's not a free market, because by law, the Federal government gets the best deal. And they define what that best deal is (ie. price floor). Just like other price-control measures implemented in the late 60's and early 70's (the 1973 oil crisis is another good example), the results are higher costs.