Jmho, but no US service person who is worthy of their uniforms would dare shoot an enemy soldier who has already given up the fight, turned and is fleeing. Not to say it doesn't happen, of course, because I'm sure it has, and does. But only a coward with a yellow streak running the entire length of his/her back would do so.
Not sure what you base this opinion on, but it is not the Law of Armed Conflict.
It depends, I guess on how you define "given up the fight," but most soldiers define that as the enemy dropping their weapons and raising their hands. "Fleeing," by its very nature is trying to move out of a position of disadvantage in order to resume fighting in more favorable conditions. Thus, you cannot general "give up the fight"
and "flee" at the same time. You can do one or the other.
To give you an example, at the Falaise Gap in July 1944, the Germans were fleeing from Allied encirclement so they could make a stand somewhere else and resume the fight. And the US Army Air Force and the US Army's VII Corps and XV Corps were killing Germans by the bushel (10,000 German KIA, to be exact). Completely legitimate.
Now, if the Germans had disabled or dropped their weapons, they would have been taken prisoner, no problem. They were
fleeing, in order to rejoin the fight elsewhere and fight from a position less disadvantageous.
At the lower tactical level, the same thing applies. In the film
Saving Private Ryan, once the US Army Rangers get to the top of the bluff overlooking Omaha, and are knocking out bunkers from the rear, the Rangers see some German soldiers attempting to run to the German rear across the open ground behind the bunkers. The Germans are carrying their weapons openly, and are fleeing from the bunkers, because the Americans have a positional advantage over them. The Rangers gun down the Germans. Fair game. No Law of Armed Conflict problem.
If they had dropped their weapons, raised their hands, and
stopped fleeing, they would have been taken prisoner.