To increase military spending without causing unrest requires something to rally around, and Trump taking over Greenland would be exactly that thing to rally around. Even though the us already controls Greenland defacto, it doesn't matter. The nationalism can be generated either way.
Only way for Europe to back down from further military spending is to first back down from foreign military adventures, then make a detente with Russia. Even then Europe will still need some manufacturing and military spending and energy spending to simply be able to hold against the US and avoid vassalisation. That's the only realistic good ending.