Important note that Greenland is not in the EU (although Denmark is), so likely the EU would at first think this is not their concern. (I think this would be a wrong thing to think, but that's how it would be diffused at first.)
No, the EU might not contain the top-of-the-top when it comes to people aware of what's going on, but not even EU is so dimwitted to brush aside an invasion of Greenland as "not our concern".
Not to mention that Greenland is an autonomous part of Denmark, and Denmark is very much a part of the EU.
So far the French president is welcoming what happened in Venezuela. How do you imagine he feels about the implications for Greenland and Europe?
https://www.politico.eu/article/france-fm-jean-noel-barrot-f...
https://www.nato.int/en/what-we-do/introduction-to-nato/coll...