The veteran singer was due to appear at Canterbury Cathedral on September 26, but his spokesman said Sir Cliff "doesn't want the event to be overshadowed by the false allegation and has therefore withdrawn".
He added that the singer was "sorry for any disappointment or inconvenience caused".
Sir Cliff's apartment in Berkshire was searched by officers from South Yorkshire and Thames Valley police last week as part of an investigation into an alleged sexual assault on a boy at a religious event in 1985.
Sir Cliff, who was in Portugal when the search took place, has firmly denied any wrongdoing.
Fellow showbusiness veteran Cilla Black has rallied to Sir Cliff's defence, saying: "I, like everyone else, was shocked to hear of these allegations and I am absolutely positive that they are without foundation."
The raid on the star's penthouse caused controversy when the BBC broke news of the search, with a film crew reportedly arriving on the scene before the police.
The corporation's director-general Tony Hall and chief constable of South Yorkshire Police David Crompton will face a grilling by MPs over the matter.