Private colleges shouldn't be taking public money to begin with. Hopefully this ends with colleges having their (private college) tax exempt status revoked. Anyone who's been to Massachusetts knows how much real estate Harvard (and MIT, and BU, and Boston College, etc, etc.) owns. Why should they be tax exempt? End it now. Also, Trump's demands are silly and not even internally consistent.
Harvard was founded to educate America's clergy. If there is a category of non-profits that could hop, skip and jump to religious exemption, it's the Ivy League.