The race row was sparked when unused footage from Top Gear was revealed in the Daily Mirror showing the TV host swinging his finger between two cars while reciting a children's nursery rhyme. In the clip, he can be heard chanting: "Eeny, meeny, miny, moe" before mumbling what sound like "catch a n***** by his toe".
Clarkson initially denied the allegations, insisting on Twitter: "I did not use the * word. Never use it."
However, after a series of meetings with BBC bosses, the presenter was forced into a humiliating u-turn.
In a video on his Twitter page, he admitted the nursery rhyme he chose to recite is well known for containing a "racist word". Clarkson insisted he was "extremely keen to avoid" saying the word, so claimed he "mumbled" where it was meant to appear.
He said he was "mortified" and "horrified" when he realised it "did appear that I'd actually used the word I was trying to obscure".
Clarkson said he contacted producers to ensure the version broadcast by the BBC in February last year included the word "teacher" instead.
Clarkson's critics said the apology was "insufficient" and demanded he be sacked. Solicitor Lawrence Davies, who runs the Equal Justice law firm, said: "This explanation is just not credible. If he really wanted to avoid saying the N-word, he could easily have avoided reciting that rhyme altogether.
"He deserves to be sacked for this. But the BBC have clearly brushed it under the carpet. "They are too afraid to lose him."
Top Gear co-star Richard Hammond said: "I can list a million things wrong with him, but he is absolutely not racist."
And James May said: "Jeremy Clarkson is not a racist. He is a monumental b******* and many other things, but not a racist. I wouldn't work with one."
A BBC spokesman said: "Jeremy Clarkson has set out the background to this regrettable episode.
"We have made it absolutely clear to him, the standards the BBC expects on air and off. We have left him in no doubt about how seriously we view this."
Jeremy Clarkson's troubled history
• April 2014: Named his black terrier Didier Dogba after the Ivory Coast footballer.
• March 2014: Joked that a bridge with an Asian man walking across it had a "slope on it''. His producer Andy Wilman claimed Clarkson did not realise it is an offensive term.
• January 2012: Made a series of derogatory remarks about India and ridiculed the sanitation for its poor by driving around slums in a Jaguar fitted with a toilet.
• January 2012: Compared synchronised swimming with the deaths of 23 Chinese cockle pickers in Morecambe Bay in 2004.
• July 2011: Said his new TV's energy-saving setting made it seem as if programmes were being "presented by Lenny Henry in a cave". Lenny Henry is a black British actor and comedian.
• February 2011: Joked about Mexicans, prompting complaint by country's ambassador.
• August 2010: Attacked by disability charities after he said the F430 Speciale car should be called "Speciale needs".
• October 2009: Accused TV producers of being fixated with having "black Muslim lesbians" on programmes.
• February 2009: Described former British Prime Minister Gordon Brown as a "one-eyed Scottish idiot''. Brown is Scottish and blind in his left eye.
• April 2007: Derided a Malaysian car he suggested was built by "jungle people who wear leaves as shoes".
• December 2005: Gave a Nazi salute while presenting a piece about German firm BMW. Said its satnav "only goes to Poland''.
