But the two sides agreed to create a working group to discuss ways to work through differences as Trump continues to call for a United States takeover of the semi-autonomous territory of Denmark.
"The group, in our view, should focus on how to address the American security concerns, while at the same time respecting the red lines of the Kingdom of Denmark," Danish Foreign Minister Lars Lokke Rasmussen told reporters after joining Greenland Foreign Minister Vivian Motzfeldt for the talks.
"We didn't manage to change the American position," Rasmussen told reporters outside the Danish embassy in Washington DC after the meeting concluded.
Trump says the strategically located and mineral-rich island is vital to US security and the United States must own it to prevent Russia or China occupying it.
"It is vital for the Golden Dome that we are building," he said in a social media post just hours before the meeting, referring to a proposed missile defence system.
Greenland and Denmark say the island is not for sale, threats of force are reckless and security concerns should be resolved among allies.
In Copenhagen, Danish Defence Minister Troels Lund Poulsen announced an increase in Denmark's "military presence and exercise activity" in the Arctic and the North Atlantic "in close co-operation with our allies".
Poulsen said at a news conference the stepped-up military presence was necessary in a security environment in which "no one can predict what will happen tomorrow".
"This means that from today and in the coming time there will be an increased military presence in and around Greenland of aircraft, ships and soldiers, including from other NATO allies," Poulsen said.
Other NATO allies were arriving in Greenland along with Danish personnel, he said.
Poulsen declined to name the other countries contributing to increased Arctic presence, saying that it is up to the allies to announce their own participation.
Earlier, Swedish Prime Minister Ulf Kristersson wrote on X that "some officers from the Swedish Armed Forces are arriving in Greenland today" as part of a group from several allied countries.
"Together, they will prepare events within the framework of the Danish exercise Operation Arctic Endurance," Kristersson said.
Two Norwegian military personnel also will be sent to Greenland to map out further co-operation with allies, the country's Defence Minister Tore O Sandvik told newspaper VG.
Germany will send 13 soldiers to Greenland on Thursday as part of a reconnaissance mission with other European countries, the German government and defence ministry said on Wednesday.
The mission, which comes at the request of Denmark, will take place from Thursday to Saturday, aiming to explore possible military contributions to bolster the security of the region, a defence ministry statement said.
This could, for example, include maritime surveillance, it added.
Trump said earlier on Wednesday on Truth Social that the NATO military alliance would become far more formidable and effective with Greenland in the hands of the US.
The foreign ministers of Denmark and Greenland had originally sought a meeting with Rubio, hoping to have a discussion among top diplomats on resolving the crisis between the two NATO allies.
But Denmark's Rasmussen said Vance - who visited a US base in Greenland in March - had also wanted to participate and that the vice president would host the meeting himself, at the White House.
with AP
