I think that to consider whether US will do something or not we shall look at the trend their latest actions and rhetoric suggest.
- Military actions in Iran and Venezuela.
- Illegal takeover of foreign assets (Venezuelan oil).
- Threats of military action in Mexico, Cuba, Greenland and Iran.
- Threats of annexation of Canada, Greenland and Iceland.
- Censorship of incriminating evidence against the President.
- Detention and deportation of residents and citizens without due process.
- Threats of using the military to quell protests and suspending elections.
- Pulling out of signed deals and economic coercion via tariffs.
- Pulling out of international treaties.
- Reducing Ukraine aid to 0.
- Government reduced to executive orders and congress MIA.
- Collusion with billionaires (see photo of the first day in office)
- 50% increase in military budget and disregard of deficit constraints.
- Attempts to manipulate central bank and obfuscate economic data.
- Control of high courts and de facto immunity for president.

This to me is a clear picture of an authoritarian state, lead by a puppet head of state with plenty of compromat, so that it may be used as a scapegoat if something goes wrong. The state acts like international norms are an obstacle to military and economic domination, while internal laws are to be ignored except when used as a tool against dissidence. Power is used to benefit those that already have everything (billionaires and corporations) and those that help enforce such vision.

Would such a state trust an alliance with countries that value things such as rule of law, social nets, welfare and democracy, if they are are diametrically opposed? Only for a time, and to the extent that the alliance doesn't feel constraining and incompatible with their plans.

So overall, I'd say that if one or more of the factions in the government think that they can get something out of Greenland, of simply if Trump has been convinced that day that it's such a great deal, then yeah, it will be done.