Based on just what the story said? Yes. Regardless of what had led up to the shooting, the threat to his life was over. The two were fleeing. He shot and wounded one and then basically executed her. That's murder and not self defense.
In California you can't shoot a fleeing robber. In Texas, you can.
Keep in mind that in all likelihood this was not the first burglary these two had committed, and had they successfully fled the scene without injury, this would not have been the last. They would have laughed about the close call and about beating up the 'old man' and promised each other to be more careful the next time.
IMO, Greer did society a favor.