Greenland is a vast country with a population of just 56,000. It was a colony of Denmark, and has voted to be largely self governing with its own Parliament. The King of Denmark is the ceremonial Head of State, Denmark sends an annual grant to assist, and Denmark has some powers over foreign affairs and defence policy. In practice the defence of Greenland rests with the US base and US forces that could be assembled in the event of a hostile attack from a joint enemy of Greenland and the US as members of NATO.
The US should make clear it has no intention of invading and occupying Greenland by force, as it is a NATO member and allied democracy of the US.
During the Second World War the US sent troops to Greenland to prevent a German invasion and occupation, when Denmark was forcefully integrated into the German Reich. The trend of Greenland politics has been to distance itself more from Danish influence and power, whilst showing little wish to become a state of the USA. An earlier attempt by the US to purchase Greenland as they had Alaska was rejected by Denmark.
The US position today seems to be based on worries about the threat to all that unsettled land and the adjacent sea routes from Russia and China which is seen as a threat to North America itself. It is also likely based on the possible exploitation of minerals, oil and gas that may lie below the ice and could be useful to the arsenal of democracy. President Trump probably wants a deal that the US offers more defence security to Greenland in return for good US access to minerals and economic development opportunities. Greenlanders may be reluctant to allow mineral and fossil fuel exploitation for environmental and sovereignty reasons.
As the US Secretary of State, the First Diplomat, has said, these matters need resolution by negotiations. The defence of Greenland is an important issue for all NATO members, and Greenland though with a tiny population might find more ways to help. The US needs to remember the purpose of NATO is to defend democracies and the rights of people to govern themselves. Maybe Greenland should start with a referendum on whether it wishes to loosen its remaining ties with Denmark as part of seeking a solution to its future.