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Greenland is an "autonomous country of the Kingdom of Denmark" with "home rule" granted since 1979. As part of a self-rule law passed in 2009 (section §21), Greenland can declare independence at any time with only a referendum of the people needed. In fact a recent poll found that 64% of the population supports this idea.
Does this mean that Greenland could declare independence at anytime and then turn around and make a deal with America, without the approval of Denmark?
Under current law, the source for a sentence saying that they can vote to get independence in the Wikipedia article you linked to says that (translated with Google, edited for grammar):
The Government {of Denmark} and {Greenland's government} cooperate in international affairs as provided for in this Chapter and look to safeguard Greenland's interests as well as the Kingdom of Denmark's overall interests.
Currently, they'd need Danish permission if they wanted to sell themselves. Now, moving on to what independence would be like:
A decision on Greenland's independence is made the Greenlandic people.
A decision shall be made in accordance with paragraph 2. 1, pre-negotiation liaises between the government and {Greenland's government} for the purpose implementation of independence for Greenland.
An agreement between {Greenland's government} and the government on implementation of independence for Greenland must be concluded with the consent of {Greenland's parliament} and must be approved by a referendum in Greenland. The agreement must also be concluded with consent of the {Danish} Parliament.
Independence for Greenland implies that Greenland takes over the supreme court over Greenland.
This makes it very clear that Denmark would negotiate with Greenland's parliament to let them go and a referendum would have to pass. Denmark's parliament also would have to vote on it. Denmark has pretty clear rules for how Greenland can leave, and unless the deal had very restrictive rules making them effectively not autonomous, nothing currently in place would stop them from dealing with other countries if they were independent.
