The environmental group, which has launched a campaign near Cairn's oil rig, said the Arctic "could be flooded with oil companies, all trying to operate in hazardous polar conditions".
It says the company is drilling two wells off the west coast of the Arctic country, and plans to drill two more before the end of October.
The region is home to blue whales, polar bears, seals and migratory birds.
"To be here and see a huge drilling rig in this beautiful and fragile environment is deeply shocking," Leila Deen, a Greenpeace activist onboard the Esperanza, which is protesting at the Cairn drilling site, said.
"This operation is too risky and companies like Cairn need to leave the Arctic alone and instead work quickly to develop safe and clean alternatives that will actually help us get off fossil fuels for good," she said.
But the exploration firm said that it complies with "some of the most stringent regulations globally" that are laid down by Greenland's government.
Three decades after an exploration effort failed to find oil, drilling in the deep ocean off Greenland's west coast resumed in 2001.
Exploration so far has been unsuccessful.
Cairn started drilling 175km west of the Disko Island on July 1 in water depths of 300-500 metres.
Although seismic data indicate that the area could have gas and oil reserves, Cairn's finding does not automatically mean that oil has been found, Erling Halfdan Stenby, a chemistry professor at Denmark's Technical University, said.
"It could be that there was only gas, it could be that there was oil too," he was quoted by the AP news agency as saying.
"On top of that, one cannot even be sure that the finding is profitable."