I don't see any evidence for that. A couple counter-examples: Singapore and developing countries like India and China have more market based healthcare systems than Western Europe and the US, and spend far less on healthcare as a percentage of GDP. India and China are making significant gains in life expectancy, but that may be explained away by the fact that they're starting from a lower level. Singapore however has a higher life expectancy than European countries.

The US is the outlier, with approximately half of healthcare spending being private, yet spending a very high percentage of GDP on healthcare. But I would argue that much of this 'private' is quasi public, as it's a manufactured outcome as a result of significant intervention by state and federal governments, like tax incentives for employers to provide healthcare through large insurance companies, and state mandates requiring health insurance to be comprehensive.

The fields of medicine in the US less subsidized via these kinds of interventions; cosmetic and laser eye surgery, have seen prices increase at below the rate of inflation.

The key is that people pay out of pocket for cosmetic and laser eye surgery:

http://healthblog.ncpa.org/why-cant-the-market-for-medical-c...

The government intervention in the private US healthcare system encourages exactly the opposite.

Also with respect to US healthcare spending, it's worth mentioning the US is a higher income country than other OECD countries, and there seems to be a trend toward healthcare spending as a percentage of GDP increasing with level of wealth, independent of all other factors. The US does have a more capitalised healthcare system, with more diagnostic equipment and advanced treatment centres for things like neonatal care. Often not mentioned is that the US has some of the best cancer survival rates in the world.

Mind you, the US is definitely suffering from the cost disease in healthcare, but the relatively higher wealth level of the US may explain the difference with Europe.