The drummer – a member of the rock band since 1963 – is said to have "passed away peacefully in a London hospital surrounded by his family".
Guitarist Wood, 74, shared a picture of himself and Watts captioned: "I love you my fellow Gemini~ I will dearly miss you ~you are the best."
Lead singer Sir Mick, 78, paid tribute to his colleague of almost 60 years by sharing a picture on social media of Watts laughing as he sat at his drumkit. He did post any words.
Guitarist Richards, 77, also chose to salute his bandmate with a simple picture – sharing a photograph of a drumkit with a 'Closed' sign hung on it.
Earlier this month, it was announced that Watts – who turned 80 in June – would not be joining the Rolling Stones on their forthcoming US tour due to an undisclosed illness.
Watch: Beatle Sir Paul McCartney is among those to pay tribute to Rolling Stone Charlie Watts
Queen's Sir Brian May, Beatle Sir Paul McCartney, and Sir Elton John were among the many other stars of the music industry to also offer their condolences to Watts' family and hail his talents as the "ultimate drummer".
Sir Brian, 74, shared fond memories of working with Watts and called him a "pillar of strength" for the Rolling Stones.
The Queen guitarist wrote: "Sad sad news in the passing of Charlie Watts. Oh Lord. Deepest sympathies to his family – and to Keith, Mick and Ronnie to whom he was certainly beloved family.
"For some people this might be a cliche - but in Charlie’s case it’s the absolute truth - he was the nicest gent you could ever meet. And such a pillar of strength for the Rolling Stones - to whom he brought a touch of Jazz and a mountain of pure Class. Bless you Charlie. Rest in Peace and Rock on. Bri"
Sir Paul, 79, was visibly shocked in the video he posted online following the news and said the news will be a "huge blow" to the Rolling Stones.
The Beatles star said: "So sad to hear about Charlie Watts. He was a lovely guy, and I knew he was ill, but I didn't know he was this ill.
"So lots of love to his family and condolences to the Stones. It will be a huge blow to them because Charlie was a rock and a fantastic drummer.
"Love you Charlie. I'll always love you. Beautiful man. Great condolences and sympathies."
Sir Elton, 74, shared a picture of himself with Watts and paid tribute not just to Watts's musical skills but also his sense of style.
He wrote: "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 drummer Sir Ringo Starr, also tweeted a picture, writing: “God bless Charlie Watts, we’re going to miss you man, peace and love to the family, Ringo.”
Watts was a key part of the Rolling Stones sound, including on classic tracks like (I Can’t Get No) Satisfaction, Start Me Up, Jumpin’ Jack Flash and Brown Sugar.
Tributes to Watts have flooded social media from celebrities and music fans alike. Others to pay their respects online include Queen drummer Roger Taylor, The Police drummer Stewart Copeland, and Canadian rocker Bryan Adams.
A statement from Watts’s London publicist Bernard Doherty to the PA news agency on Tuesday said: “It is with immense sadness that we announce the death of our beloved Charlie Watts. He passed away peacefully in a London hospital earlier today surrounded by his family.
“Charlie was a cherished husband, father and grandfather and also, as a member of The Rolling Stones, one of the greatest drummers of his generation.
“We kindly request that the privacy of his family, band members and close friends is respected at this difficult time.’’