If you own a place, you can say who goes in and who doesn't (short of being racist, which while I don't condone any kind of rascism, I still think the owner should be able to not allow anyone he wants for whatever reason he wants). It is a private establishment, of which the owner may or may not choose to open to the general public, meaning to allow anyone and everyone in. That doesn't mean it becomes public property.

Let's say that you open your home to the general public, meaning you'll allow anyone who wants to come in to come in (let's say you're having a garage sale or something). I decide to go in your house and I am allergic to cats (they'll get me really sick). I go in your house and see you have cats, now do I have a right to legally have your cats removed so that I can enter your home, which you so graciously invited me to do? Afterall, it's not illegal nor wrong to own a cat, yet they have a negative effect on my health. Yes or no?