Thursday's show was intended to be the culmination of a European tour which has taken the musician through 30 cities.
However, the 61-year-old guitarist yesterday released a statement on his website announcing the concert had been cancelled.
It said: "Eric Clapton is extremely sorry to disappoint his Russian fans but the circumstances of the cancellation are completely beyond his control."
There was speculation in Moscow that Clapton's management had cancelled the show because they were unhappy that the venue was an open area that adjoins Red Square - rather than the landmark itself.
Russian fans had paid up to 200 to hear Clapton perform. But the Russian tour promoter has suspended ticket sales and begun offering refunds.
Pavel Grinshpun, the editor of the Russian edition of Rolling Stone magazine, said there was speculation that the real reason for the cancellation of Clapton's Moscow show was the location.
"The story is that Eric Clapton's management weren't satisfied with the venue they were going to play because it is not actually Red Square.
"One problem is that Red Square is not only a square, it is a cemetery as well with the graves of Lenin, Stalin and other people from the Soviet era - it is always controversial for the authorities to let musicians play there."
The Kremlin must give permission to play in Red Square as the historic site, and landmarks such as St Basil's Cathedral, are classified as Russian Federation property rather than belonging to the city of Moscow. A spokeswoman for the Kremlin's press service said she could not comment on Clapton's cancellation.
Both Moscow and St Petersburg are routine stops on the touring schedules of major acts. Red Square has been the scene of performances from artists including the former Beatle Sir Paul McCartney, and the classical singers Placido Domingo and Luciano Pavarotti.
However, the majority of acts have played in an open area called Vasilyevsky Spusk, which adjoins St Basil's Cathedral but is very close to Red Square. It was the venue for last year's Live8 concert in Moscow and also played host to Roger Waters in June when the Pink Floyd star played the entire Dark Side of the Moon album live. Other acts who played recently in Vasilyevsky Spusk include Shakira and the Black Eyed Peas.
The performance would have been the first time that Clapton had played in Red Square and the guitarist was said to be delighted at the prospect.
InterMedia, the Russian tour organiser, was reportedly preparing to release more details today about why the Clapton show fell through.
Clapton is the second big rock act to cancel in Russia this year. The Rolling Stones called off a June concert in St Petersburg, part of the band's A Bigger Bang world tour, to let guitarist Keith Richards, 62, recover from a head injury suffered in a fall.