No, that's *not* how it works
If you want to open a business to the public, you have to open it to *all* the public or *none* of the public. That's how the rules work. Because the simple fact is, "free market" wanking notwithstanding, civilisation is about protecting the weak from the worst excesses of the strong.
The glibertarian line -- that everyone is free just to start their own, alternative business which, by virtue of its non-exclusionary nature, will end up being more successful than Google -- turns out in practice to be bollocks.
Business owners are by definition in a position where they have the upper hand over their customers -- and it is in everyone's interests that they do not abuse that unequal relationship by capriciously excluding entire sectors of society.
Try opening a whites-only hotel and see how long you last -- even if you place a stack of brochures for inexpensive surgical skin colour reassignment in an area not subject to the colour bar.