It is private property, but it is legally a public accomodation. You are not allowed to discriminate in service, and if you have any requirements of dress and deportment ("no shoes, no shirt, no service" etc.) you have to post them.
You cannot just randomly pick out someone who is not harassing your customers or staff and throw them out.
The proper thing to do if the patron will not leave would be to call the police.
"You cannot just randomly pick out someone who is not harassing your customers or staff and throw them out."
Then essentially, it's not your business. If somebody is taking up space in my place reserved for a paying customer, I'll give them the boot. I'm in business to make money, not out of some obligation or agreement to provide some "public accomodation". It's my credit, my money, and my time invested, not the government's.
"No shoes, no shirt," etc. is a health concern and need not be posted anywhere that I'm aware of. If a health inspector walks in your restaurant and sees a customer without shoes, you are likely getting hit with a violation.
None of that is the issue here, though. The issue is some goof writing a law because some guy got thrown out of a bar.