From Tuchman's The Guns of August:
A question that [Brigadier General Henry] Wilson asked of [General Ferdinand] Foch during his second visit in January 1910, evoked an answer which expressed in one sentence the problem of the alliance with England, as the French saw it. “What is the smallest British military force that would be of any practical assistance to you?” Wilson asked. Like a rapier flash came Foch’s reply, “A single British soldier—and we will see to it that he is killed.”
People's attitudes often change when their sons and daughters are on the line.
I don't know what the original source meant by it, but I interpret it as being a tripwire force. It works by the invading army killing the deployed forces.
War is also about internal politics. Do you really think someone can invade and potentially kill the military of your allies without repercussions? But let's ignore that and game out the next steps.
So some countries are pissed off for them killing military assets but the invading army think they wouldn't escalate? That's fine. Let's continue from the scenario where this happened.
Let's talk about killing civilians. You don't think that taking over Greenland will be without civilian deaths? Even if that's ok then let's continu to the next point:
Greenland is now part of the USA, this means border protection. This means you need to import border patrol or ICE or whatever into Greenland. How do you think that would work out? Non-military persons overseeing deporting non-greenland people in the artic. The US doesn't have that much equipment or people trained for artic conditions. Most of your guns, tanks, etc... don't work there reliably. Anyways...
Do you want the military overseeing deporting non-greenland people from your now US territory? Even when it's the most peaceful takeover ever in the history of humankind, it's still going to be a mess and people will die.
I've given you the most optimistic situation: Canada will allow overflights to take over Greenland. Greenland won't resist. The EU will do nothing even when their people are killed. etc...
To return to your question, killing an allied soldier will have some effects in the internal politics of the allies of the US.
People's attitudes often change when their sons and daughters are on the line.