I don't see why that same thing couldn't apply for healthcare. 1. ERs should publish prices in a public database. If a price is not published for an item/procedure, the customer can pay the lowest price she/he can find for the same procedure from an other provider. 2. The same price must be charged from everyone (regardless of insurance).
Even if only a small fraction of people shop around it might push down the prices. Even if the above is too naive, I am sure someone smarter can come up with a less naive version of it.