> Shouldn't it be Walmart's right to do what they want on their property, and my right to decide not to visit Walmart if I don't agree with that. Isn't the converse an infringement of Walmart's rights?

No. Property "rights" are secondary to human rights. Like, Walmart can't knowingly sell poison as food just because it's their property...

In your example, Walmart is free to record you on security cameras for security / theft purposes. However, they can't record what you're looking at and reuse that information for targeted advertising without consent - profling is simply not required to do business, so your right not to be profiled wins.

What law prevents them from doing this?

There's an asymmetry between what a company can do politically/legally given it's resources and what an individual can do. This is why countries generally have some kind of laws protecting consumers.

Security cameras recording footage and it not being used in 99.999% of the time when no crime occurs is fine. The tapes aren't kept forever. Just as having server logs to identify malicious actors i.e. hackers or scammers is fine. What's not fine is e.g. running facial recognition on the security camera footage, or figuring out who bought what (cough Amazon Go).

"Tesco is set to install hi-tech screens that scan customers' faces in petrol stations so that advertisements can be tailored to suit them, it has been reported.

The retailer will introduce the OptimEyes screen, developed by Lord Sugar's Amscreen, to all 450 of its UK petrol stations, in a five-year deal, according to The Grocer.

The screen, positioned at the till, scans the eyes of customers to determine age and gender, and then runs tailored advertisements."

https://www.theguardian.com/business/2013/nov/03/privacy-tes...