No, there will be no invasion and Denmakr and France are not going to war with the US.
Denmark, a colonial power, starting to give Greenlanders rights in 1953, The US had been interested in buying the colony even before that date.
OPTIONS
1) Greatly increase the numbers at the current base and start exploratory mining near the base. This is allowable under the current arrangemnets.
2) Offer Greenlanders $1M each and call for a vote, allowing the same self-government that they have now. The US would be responsible for defense, foreign policy and mineral development. The first two are the RESPONSIBILTY of Denmark now (not that they do anything),
3) Negotiate a treaty with Denmark and Greenland. I don't see how Denmark would be involved after such a treaty, but that would be negotiated. They do nothing now.
I think a treaty is indeed important. Denmark was a strong ally in recent wars and should be treated as such.
4) Negotiate the sale with Greenlanders retaining most of the rights that they have now PLUS an annual stipend from the sale of minerals.