I’m not sure it is in their best interest to do so. I haven’t studied this topic seriously, but it would seem to me that health insurance is a highly inelastic product. Almost everyone who has the means to purchase it does purchase it, regardless of the cost. Because of this, the insurance companies can (and do) easily pass along price increases of medical costs right to the consumers through increased premiums.
Also keep in mind that the employer-provided system means that most Americans not only avoid paying the true cost of their medical procedures, but they also avoid paying the true cost of their health insurance, as their insurance is often highly subsidized if not provided entirely by the employer. While those of us who understand economics might realize that greater health benefits must translate to lower wages (or lesser benefits of another type), the average person seems to have not made this connection.
If the end customers don’t care about price hikes, nobody else along the chain will either.