No, I would not label this event private. Yet you would apparently label no event as private, with your continued references to supermarkets, resteraunts, and the like. You can't frame the debate as bring about publicly-funded events and then use examples of privately-run ones for your arguments.

Anyway, in most areas, I actually am ok with "discrimination." What is morally reprehensible about running an event designed to disciple someone in their religion, then asking people of another religion to stay away? I can't imagine a Muslim would be too happy about an atheist showing up at a Muslim retreat and yelling obscenities to Allah. Nor would I be terribly thrilled if I was on a men's campout and we were forced to bring women along. Like it or not, we aren't all the same, and sometimes we need to hang out with people like us in a certain area. As long as you aren't terrorizing a certain group of people out of hatred, private property laws should allow you to decide who comes on your property.

If the government pays for an event, that's another matter entirely, but please choose which debate we are having and use relevant examples so I can be clear going forward.