The Indigenous star, who was nominated for an Academy Award for Best Supporting Actor in 1991, passed away on Monday in a Toronto hospital, his representative confirmed.
Greene enjoyed a decades-long career across film, television and stage, often praised for breaking barriers for Indigenous actors in Hollywood.
His representative told Deadline: "He was a great man of morals, ethics and character and will be eternally missed."
Greene, who was of Oneida First Nation and Canadian descent, grew up on the Six Nations Reserve in Ontario and worked as a welder, draftsman and steelworker before turning to acting.
His television debut came in 1979 with a part in The Great Detective, and his first film role followed in 1983's Running Brave.
Greene's career breakthrough arrived in 1990 with Kevin Costner's epic Dances With Wolves, in which his portrayal of the Sioux medicine man Kicking Bird earned him an Academy Award nomination.
In a past interview, the actor recalled the challenges of filming, including being thrown from a horse during production.
"I wanted to get payback on the animal rather than take a break from filming," he said.
He went on to appear in a wide range of roles, from comedy such as his recurring part in The Red Green Show to dramas including The Green Mile (1999), in which he played Arlen Bitterbuck opposite Tom Hanks.
Other credits included Maverick (1994), Die Hard with a Vengeance (1995), Transamerica (2005), The Twilight Saga: New Moon (2009) and Aaron Sorkin's Molly's Game (2017).
Greene's accolades included a Grammy, a Gemini and a Canadian Screen Award, as well as a nomination for an Independent Spirit Award.
Actor Lou Diamond Phillips led tributes to the star, describing Greene as a brother.
Greene's last completed project is the thriller Ice Fall, which is scheduled for release on October 16.
Greene is survived by wife Hilary, their daughter Lilly Lazare-Greene and grandson Tarlo.
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