Not necessarily. Such an event would be a geopolitical reenactment of the Danish folk tale "The Emperor's New Clothes". Turns out the emperor isn't just naked he's still living in his parent's basement. And the emperor here is Europe and the parents are the US. Why would I as a German support a war over an island that has not just been criminally neglected by a supposed alley (Denmark), it is even politically and economically irrelevant (as of now). And not just that ... "we" have been begging the US to keep protecting us and for them Greenland is in that respect very much serving a purpose in line with that very military support. Europe is naked as it stands. I don't like it. But that's just how it is.
I find your analogy to be extremely strange. We are extremely well-armed. Greenland has also not been neglected in any way.
With regard to why you should care about Greenland, you've signed an agreement to do it, you are after all in NATO.
There has indeed been co-operation in protecting Greenland and this is very reasonable considering that the US has a presence there in the form of their military bases. I don't see this co-operation as begging.
The disrespectful, colonialist treatment of Greenland by Denmark is sufficiently well documented. And if you think Europe stands a chance in a war against US then I don't even know what to say anymore ...
No one is calling for war, but it's not like that documented history happened in this century. Greenland had been part of Denmark since the vikings. Surely we can get past the history and talk in today's terms. The people of Greenland have a voice in Danish politics and both the people and politicians said NO to wanting to have US rule them and YES to stay with Denmark.
So yes, defending Greenland becomes a case of helping a people stay free and not invaded, no matter the enemy.
It's silly to say "well we have no chance against" because then you can end that with China, Russia or even India.
Then you are the deluded one, because Europe has nukes and if Europe dissapears under US nukes, California turns into a big crater and NYC ceases to exist. Everyone - I repeat, everyone - losses. USA, Europe, China, the rest of the world. Everything lost like the Bronze Age, but far worse.
