Of course, the Constitution never mentions such a separation, but the liberals are probably right in thinking that the federal courts would regard tuition vouchers as a violation. The other liberal objection is the prediction that school choice would destroy the public school system, as most parents would naturally choose to send their children to private schools. That shows how much confidence the National Education Association and other liberals really have in the public school system they control. Apparently even they understand that responsible parents would jump at an opportunity to send their children to an alternative private school. There is a third view of the school choice issue that is seldom heard.
Private schools are superior precisely because they are privately funded and privately controlled.
The effect of tuition vouchers would be to make private schools more like public ones. Federal or state funding always comes attached to federal or state control. Private schools now pretty much control their own curricula, book selection, discipline policies, etc. When a private school accepts tuition vouchers from government, it is only a matter of time before it becomes indistinguishable from a government school. They will lose control over all of those factors that made them different and better.