Maybe. Maybe not. They arrested her boyfriend and charged him with murder, but have not arrested Greer.
If I were on his jury, I would vote for acquittal.
If you were on the jury, how would you vote? Knowing the facts released so far, would you convict him?
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.