Most parking lots are NOT owned by the stores, but by property managment companies leasing the store spaces. So even if it is private property, the store has no legal grounds outside of the space it rents.
Though, Wallmart does tend to own the entire property.
Although none of this really applies in the case, because the woman was stopped right outside the door which IS most definitely part of a stores premises.
1. Wal Marts owns their parking lots.
2. If a store confronted a person(asked them to stop) while inside the store, and the person left, they can follow and still detain.
3. It has to be on or near the premises.
They have to have a reasonable suspicion of theft. The pursuit has to be immediate. And any force used has to be reasonable. They aren't allowed to search without consent. Detainment can only be for a short time, typically 15 minutes.
Also, for those saying the old lady assaulted the black woman are wrong. It doesn't appear she grabbed the lady in the video. It looks like she touched her on the shoulder to get the woman's attention. Touching someone on their shoulder to get their attention is not civil or criminal assault.
I saw Wal Mart stop two people stealing meat. The people who were stealing were drunk and tried to fight when they were stopped. The people were taken down hard and physically detained. Cops showed up arrested the two people and the two people ended up getting charged with 1 felony and 3 misdemeanors for $60 in steaks.