Sidney died at his Las Vegas home on Sunday, after succumbing to complications of lymphoma.
He was responsible for a string of classic musicals including Annie Get Your Gun, Anchors Aweigh and Kiss Me Kate.
He made 28 feature films in less than three decades and worked with many famous Hollywood names, including Gene Kelly, Frank Sinatra, Tony Curtis, Lana Turner, Dick Van Dyke and Elvis Presley.
Sidney was born in Long Island, New York and began his show business career as a child actor on the vaudeville stage.
His film-making career began with shorts but he started to direct feature-length films for MGM studios with 1941's comedy Free and Easy.
His first musical was Thousands Cheer in 1943, starring Kathryn Grayson and Gene Kelly.
Hit
Anchors Aweigh, released in 1945, which starred Frank Sinatra and Gene Kelly as sailors on leave, received five Oscar nominations including best picture.
In 1950, he took over Annie Get Your Gun - stalled by the illness of its star Judy Garland - and completed the film with Betty Hutton in the title role.
It was a major hit, as was his 1951 remake of Show Boat and his 1953 film version of Cole Porter's musical Kiss Me Kate.
He also directed Elvis Presley in 1963's Viva Las Vegas.
Sidney's last film was the 1968 British musical Half a Sixpence, starring Tommy Steele.
Sidney was also active in the Screen Directors Guild - now the Directors Guild of America - and was its president for 16 years, receiving its first president's award in 1998.
The guild's current president Martha Coolidge said in a statement: "The Directors Guild is extremely saddened by the passing of our former president, George Sidney.
"He will be greatly missed."
Stars
In a 1998 interview, the Sidney blamed the decline of the Hollywood musical on the changing economics of the film business.
"Musicals became too expensive.
"When we were working at MGM, there was a whole company of talent - stars, directors, choreographers, song writers, conductors, arrangers.
"All worked under contract at regular salaries.
"To gather people like that today would be enormously costly," he said.
A private memorial is being planned for the director.