Theoretically. Very few people wrapping themselves in the libertarian flag, especially the "Tea Party" version are anything near close to a real libertarian.
Besides, not caring at all what other people do extends to letting them put whatever signs they feel like on their business. So libertarians are completely okay with "No gays/Jews/n*****s/atheists" signs.
Well I guess that some people see running a business as something much more than a pure financial exchange. They think selling something to a gay person is helping them be gay and encouraging people to be gay. If it was up to them then no one would be gay in their community - they would either be killed, exiled, or have to put on a very good act and not be found out their entire lives - or just join the clergy. Ditto "colored people" would be rounded up and forced into slavery. Since most of those things are illegal now then putting a sign on their door is their own pathetic protest and they don't really care if the lose at least 10% of all business, if not much more.
I like that idea. Reminds me of the people that bought a house across from Westboro Baptist Church and painted it with gay colours, put up a gay flag etc. Good for them and great way to counter hate speech.
But in general I still disagree about business open to the public (vs. private clubs etc.). The ability to run a business and be taxed advantageously is something that communities grant them in exchange for reciprocal favours. One is accommodation. Now if a community wants to change those rules fair enough. It could even say so whatever you want but with no favours from us. But let's see how many business want to keep running if they get zero favours from the government. We'll just go back to individuals working for themselves and not collaborating on anything. Of course now we have gone to the extreme of corporations and their owners running the country so that ain't ever going to happen short of mass rebellion by we the people, and a very large number are to dumbed down and brainwashed to ever consider doing anything their corporate overlords don't bless.