Mr Livingstone made his attack during an address to schoolchildren in a debate on the Iraq war.
He said he would get as much pleasure from Mr Bush being forced out of office as he had done from the downfall of former Iraq leader Saddam Hussein.
The outburst was immediately criticised by London politicians who fear a negative effect on the mayor's efforts to attract American tourists here.
Mr Livingstone was answering questions on the Iraq war and other subjects during a two-hour meeting with 200 schoolchildren at City Hall on Thursday morning.
After making a pointed reference to Mr Bush, he was asked by Channel 4 broadcaster Krishnan Guru-Murthy, who was chairing the meeting, to explain his making a personal attack on the US president when he disliked answering personal questions himself.
The mayor said: "I think George Bush is the most corrupt American president since Harding in the Twenties.