Taking our eye off of healthcare for a moment, Andrew Sullivan calls attention to the analogous case of school curricula, as reported by Rachel Tabachnik:

Some Louisiana students receiving publicly funded vouchers and attending private schools in 2012-2013 will be taught from educational media promoting young earth creationism, global warming denial, history that is not factual, and bigotry toward Catholicism, Mormonism, other Protestants, and non-Christian religions. This is predictable because some of the schools that are on the approved list to receive voucher students use curriculum from A Beka Books, Bob Jones University Press, and Accelerated Christian Education (ACE). Public funding of the teaching of creationism is already happening in Pennsylvania, Florida, and other states with "private school choice" programs.

She does the digging here, here, and here.As more private institutions take on the burden of providing services that the State deems necessary for the flourishing of its citizenry, I predict that we will have more clashes between institutional autonomy and government mandates. The fact of the matter is that you cannot take government money and use it for evangelical purposes. That's establishment.So, as public schools are abandoned and public money is funneled into "private" institutions, they lose their right to teach whatever they want. Similarly, in the absence of a single-payer system a deprivatized healthcare system, if medicare and medicaid funding, not to mention federal student aid, is used to support the heath provisions offered by "private" hospitals and universities, these institutions can no longer claim exemption from government oversight.If you want to be exempt from the mandates of the government, do not take government money. Just ask the Church of England, who fears that, because of their established status, they will be legally bound to bless same-sex marriages. Interestingly, then, in England peoplemight be coming around to the virtues of disestablishment, just as the country that invented it seems to be having doubts.