The musician, who marked his 80th birthday in June, had been a member of the iconic rock group since 1963.
A statement by his publicist said Watts was a “cherished husband, father and grandfather” and “one of the greatest drummers of his generation”.
The news comes just weeks after it was announced that Watts was to miss the Stones’ forthcoming US tour as he recovered from an unspecified medical procedure.
After the news broke on Tuesday, a host of A-list stars took to social media to pay their respects.
Sir Elton posted a picture of the pair and said in a Twitter post: “A very sad day. Charlie Watts was the ultimate drummer.
“The most stylish of men, and such brilliant company. My deepest condolences to Shirley, Seraphina and Charlotte. And of course, The Rolling Stones.”
Beatles legend Sir Paul McCartney paid tribute to Watts with a video posted on Twitter.
He said: “So sad to hear about Charlie Watts, Stones drummer, dying. He was a lovely guy.
“I knew he was ill, but I didn’t know he was this ill, so lots of love to his family, his wife and kids and his extended family, and condolences to the Stones, it’ll be a huge blow to them because Charlie was a rock, and a fantastic drummer, steady as a rock.
“Love you Charlie, I’ve always loved you, beautiful man, and great condolences and sympathies to his family.”
Sir Paul’s Beatles bandmate Sir Ringo also tweeted a picture, writing: “God bless Charlie Watts, we’re going to miss you man, peace and love to the family, Ringo.”
Singer Sheryl Crow said there was a “gaping hole in the universe” following his death in a tribute on Twitter, while Nile Rodgers tweeted: “You are a smooth brother. Thanks for all the great music.”
Nile Rodgers paid tribute to Watts, writing: “Rest In Power #CharlieWatts RIP. You are a smooth brother. Thanks for all the great music.”
Robbie Robertson, former lead guitarist and songwriter for The Band, tweeted: “Charlie’s drumming is powerful and unique. His approach is entirely his own and helped shape the sound of rock and roll. Blessings Charlie Watts.”
Musician Bryan Adams said on Twitter: “RIP Charlie Watts, one of the greatest rock drummers ever and a real gentleman.
“Condolences to his family and the band.”
Duran Duran also paid tribute to Watts on Twitter, writing: “So saddened to hear about the passing of Charlie Watts. An absolute inspiration to a legion of drummers since the 1960s. A man of grace, style, dignity and composure.”
There were also tributes from outside of the music sphere, with Edinburgh author Irvine Welsh writing on Twitter: “RIP Charlie Watts.
“Always steadfastly came over as a no-nonsense honest broker in a situation where the acclaim, wealth, pomposity and egotism could corrupt the sweetest of souls.”