In OTL the indigenous tribes of Greenland were able push forward Homerule in Greenland by 1978. Both by peacefull and not peacefull means.
But what if the danish Government had gone the other way? Send troops to Greenland. For instance fearing that the USSR would send weapons to the tribes up there in a way to force the Americans out of Thule?
Could this have caused a war?
What "tribes", what "push forward" and what "not peaceful means" in OTL?
I find it very hard to recognise your description of Greenland and the OTL events there. If we shall approach a situation like the one you describe we need a serious PoD in Denmark - like an ultra-nationalist government insisting on spending billions of kroner each year to keep Greenland from sinking into poverty(half the public spending in Grenland comes from Copenhagn). And if that is the PoD, I think we would have bigger and more interesting events than anything on Greenland!
Anyway with only about 50.000 inhabitors, most nowadays living in towns, a rebellion is faily easy to crush - or ignore. There is no way the Greenlanders can force USA out of Thule. Thule airbase is situated in one of the most deserted places on earth - only a few hundred Greenlanders ever lived anywhere near, and most if not all were moved south when the base was built after WWII.
The base was and is beyond discussion as it is instrumental in keeping a trustworthy "balance of terror" giving early warning of an ICBM attack on USA over the North Pole. USSR trying to stir the pot around Thule would have the Cuba crisis be diplomatic sense and require stupidity beyond philosofical imagination in the Soviet leadership. I believe the Greenland left-wing (IA) from time to time has demanded "USA out of Greenland" but only symbolically and to my knowledge with a remarkable distance to Moscow. I just checked their web site and couldn't find any present issues at all about Thule or USA.
I think the closest you can get would be an incident like at Klaksvik on Faroe Islands in early 50's. Here armed police was sent from Denmark to enforce the sacking of an ex-nazi but locally popular doctor. Thank God all parts came to sense before fire was opened. I doubt if any government not put in power by ASBs could survive killing people to stay in power (and paying) in Greenland or Faroe Islands - the only issue would be securing base rights for USA/NATO, but that could be arranged and be a comfortable source of income for the new independent micro nations.
Regards
Steffen Redbeard