Johnson, who was forced to make a high-profile apology last year for a Spectator article in which he criticised Liverpudlians for wallowing in grief after the murder of Ken Bigley in Iraq, wrote his comments in his Daily Telegraph column today.
The Conservative MP and Spectator editor makes an unlikely supporter of Livingstone, who is known for his left-wing policies.
Johnson calls the row a "non-story, a media spat" and writes: "I do not normally side with Red Ken, but on this occasion I say, Ken, whatever you do, don't apologise. Tell the papers to take a running jump and tell Blair to join them."
At a press conference on Tuesday, Livingstone said he would not apologise because he had no need to.
"You can make the case that my remarks were offensive and they may be actionable but you can't make the case they were racist. I am not going to apologise if I do not believe I have done something wrong. I am not going to appease media pressure by lying," he said.
"I could apologise but why should I say words I do not believe in my heart? Therefore I cannot."
Speaking on Five's 'Wright Stuff' show yesterday, the prime minister added his voice to those calling for Livingstone to say sorry.
"Let's just apologise and move on," Blair said.
Jewish groups and the culture and media secretary, Tessa Jowell, have also asked the mayor to apologise to the Jewish reporter to whom he made the comment, which has been labelled anti-Semitic.
The two exchanged words following a party marking 20 years since Labour's Chris Smith, who recently revealed he was HIV positive, came out as the first gay MP.
Livingstone is said to have told the reporter, Oliver Finegold, that he should work for a paper "that doesn't have a record of supporting fascism".
The comment was a reference to the Standard's sister paper, the Daily Mail, which in 1934 ran a front-page story declaring "Hurrah for the Blackshirts".
The Evening Standard, which splashed the story, quoted Auschwitz, Buchenwald and Bergen-Belsen death camp survivor Gina Turgel MBE as saying "perhaps the mayor does not understand the pain we went through".
Henry Grunwald, president of the Board of Deputies of British Jews, told the paper that Livingstone should consider his position.
"These words are appalling. His insensitivity seems to know no bounds. He should consider his position as mayor of this great city," Grunwald said.
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