There's a difference between private property and private property that's open to the public. The mall is private property but it is open to the public, which means that you cannot restrict access to any particular person based on grounds that are protected by the Constitution. For example, the mall can't say that Blacks, Gays, reporters, feminists, priests, the handicapped, etc., are barred from entering the mall. Likewise, it's also unconstitutional to say that someone bearing a firearm cannot enter, as this is a constitutionally protected right. There is no right to go about naked, so they can impose reasonable restrictions on dress.
The line between the rights of property owners in the case of closed and public access property were clearly defined when the Jim Crow laws were struck down. So what bothers me is that people who would have a cow if they were told they couldn't go into the mall because they're black, or gay, or in a wheelchair don't seem to mind if someone is told they can't enter because they bear a firearm, even though the law is just as unconstitutional.
The fact that this law was allowed to be passed and has stood without challenge bugs the crap out of me, and I'm dang glad this guy is challenging it. What also bugs me is that this guy will get no help from either the ACLU or the NRA.