The studio confirmed that Calley passed away in LA after a long illness, but did not specify the ailment.
Calley worked as an executive for Warner Bros and MGM for many years, before joining Sony as its chairman and chief executive.
He oversaw the production of many memorable films, including Spider-Man, Angels & Demons, Closer and Postcards from the Edge.
The Academy of Motion Picture Arts & Sciences gave Calley its Irving G Thalberg Memorial Award for his contributions to the film industry in 2009.
The Exorcist director William Friedkin paid tribute to the film executive, telling The Hollywood Reporter that Calley played a key role in guiding his career.
Friedkin said: "John Calley was the very best film executive I've ever worked with. His comments and his support of The Exorcist made the film what it became.
"I'm forever grateful to him, I miss him and the industry has not seen anyone comparable since his retirement. He was a fine man as well as a fine filmmaker."
Actress Maria Bello revealed that she was planning to visit Calley on the day that he died.
Bello explained: "I was supposed to meet him at 9am and say goodbye, [but] he died at 4 in the morning. He was struggling in the last couple of years, but I have faith that he is playing jazz and dancing around with a lot of beautiful women somewhere…
"He was an inspiration to so many people because he treated others with dignity and a sense of humour and knew they were perfect and lovely the way they were. He was a teacher in so many ways, incredibly human and incredibly spiritual."
Calley is survived by a daughter and three step-children from his seven-year marriage to The Big Chill actress Meg Tilly.
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