He was widely regarded as one of the country's finest contemporary classical actors, with a long association with the company.
Its artistic director and Sir Antony's husband, Gregory Doran, had taken compassionate leave to care for him.
The company said it was "deeply saddened" by the news.
RSC chair Shriti Vadera said the actor was "beloved" in the organisation "and touched and enriched the lives of so many people".
"Our thoughts and sincere condolences are with Greg, and with Antony's family and their friends at this devastating time," added Catherine Mallyon, RSC executive director and Erica Whyman, acting artistic director.
Sir Antony's film appearances included Shakespeare in Love and Mrs Brown, while his RSC credits included Richard III and Macbeth. He also appeared in TV series including The History Man and the BBC's Murphy's Law.
Once described by Prince Charles as his favourite actor, Sir Antony played many of the great Shakespearean roles, from King Lear to Shylock.
But it was his portrayal of Richard III as a villainous hunchback on crutches which won him an Olivier Award in 1985.