After chasing the robber into a corner of a parking lot, the robber pulled a gun on the permit holder who then drew, shot and killed the thief.
I reckon it comes down to how much right one has to recover property when stolen, short of committing a homicide. That won't be allowed in any U.S.jurisdiction outside of Texas.
But when the robber pulled out a gun, that was the start of a new crime, and the shooter's response was in self defense. Unless the cops simply wrote the whole thing up as an armed robbery come to a favorable conclusion.