Drummer Watts, who became part of the Stones’ longtime foursome alongside Jagger, Richards and Wood, played for the band for more than 50 years.
His bandmates led the tributes on social media, with Jagger posting a picture on Twitter and Instagram of Watts drumming.
Richards also posted a picture to his social media accounts of a drum kit with a “Closed” sign hanging over it.
Wood posted a picture of himself with Watts, adding: “I love you my fellow Gemini, I will dearly miss you. You are the best.”
The Stones had announced earlier this month that Watts would miss the band’s upcoming North American leg of its “No Filter” tour after undergoing a medical procedure for an unknown condition.
On Tuesday, Watts’ spokesperson told CNN Watts had “passed away peacefully in a London hospital earlier today surrounded by his family,” adding that he was “one of the greatest drummers of his generation.”
Several other musicians also shared their memories and tributes for Watts on social media, with many saying they had been personally inspired by the drummer.
In 1962, Jones formed the Rolling Stones with singer Jagger, pianist Ian Stewart and guitarists Richards and Dick Taylor. Watts initially turned down the group’s first offer to join, later conceding and playing his first gig with them in January 1963.
Watts is survived by his wife of more than 50 years, Shirley, his daughter Seraphina, and his granddaughter Charlotte.
CNN’s Hilary Clarke contributed to this report.