How will the EU get this arctic warfare equipment to Greenland? If the EU is so ready and willing to use this equipment, maybe they should deploy it to Ukraine instead.
The US literally has bases in Alaska and Greenland and deals with these temperatures regularly.
Sure if the US decides it would like to leave Greenland? It's just depending on who wants what with what influence factors. But if both blocks just decided to put all of their might and resources and politics capital into this the US clearly can just take Greenland and there's nothing the EU can do about it.
Most regular equipment doesn't work in -40C temperatures, to take Greenland USA would need to develop more arctic enabled weaponry just for that.