A key issue is the element of possession.
It’s one thing if you are arrested for something that wouldn’t be a crime in America over in say, Singapore. I could see America’s State department trying to step in if we had someone facing death for drug smuggling, but Singapore has a lot of other crimes we don’t have, most of those, you will face the punishment as required under the laws of Singapore.
You committed an offense in their jurisdiction, and most importantly “they’ve got you.” That isn’t even really a matter that concerns the U.S. court system.
However, if they charge you with something that isn’t a crime in the United States, and you escape to the United States, we are not going to extradite. We won’t extradite people for things that aren’t also a crime in the United States. Namely because it is explicitly noted in all our extradition treaties, but it is explicitly noted because no American court would extradite an American citizen to some crazy foreign country that wants to chop their hands off for something that isn’t even a misdemeanor in the United States.
That isn’t a perfect analogy to this situation because the soldier involved committed an offense that is certainly a crime in the United States. However, I can also say it is highly unlikely an American court would ever extradite an American citizen to stand trial in countries like Aghanistan, Iran, Turkmenistan, Myanmar, North Korea etc. Even if I go into North Korea, steal a bunch of priceless gems from the Dear Leader’s palace, and escape to America, no America court is going to extradite me. Not because my action wasn’t criminal, but because no American court will subject an American citizen to the court system of a country like North Korea (we also do not have a formal extradition treaty, so there would be nothing to compel our executive branch to even try to extradite, and they wouldn’t.)
We have possession of this guy, we can’t morally give him over to a court system where we do not believe he will receive a fair trial. It would be an immoral act. It would be a different matter if we didn’t have possession of him, but we do. It’d be a different matter if we were talking about turning him over to Germany, Japan, South Korea, or some other country that has a criminal justice system where you can expect a fair hearing and due process. Not the same sort of legal process as you’d find in America, but you still have rights and protections.
Just because this man has acted immorally that does not excuse us acting immorally towards him.