The 62-year-old TV presenter said in his column that he dreamed of the duchess being paraded through British towns and publicly shamed.
Clarkson said his language was “disgraceful” and he is “profoundly sorry”, but the duke and duchess said he did not address “his long-standing pattern of writing articles that spread hate rhetoric, dangerous conspiracy theories and misogyny”.
The Sun later apologised and said it regretted the publication of the column, which last week surpassed more than 25,000 Ipso complaints.
Prince Harry branded the article about his wife “horrific, hurtful and cruel” during an interview with ITV’s Tom Bradby to discuss his autobiography Spare, adding that what Clarkson had written would encourage people around the world to believe it is an acceptable way to treat women.
In a lengthy statement on his verified Instagram account, The Grand Tour presenter said he emailed the couple on Christmas Day to say his language in the column had been “disgraceful” and he was “profoundly sorry”.
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“One of the strange things I’ve noticed in recent times is that whenever an MP or a well-known person is asked to apologise for something, no matter how heartfelt or profound that apology may be, it’s never enough for the people who called for it in the first place," he said.
“So I’m going to try and buck the trend this morning with an apology for the things I said in a Sun column recently about Meghan Markle. I really am sorry. All the way from the balls of my feet to the follicles on my head. This is me putting my hands up. Its a mea culpa with bells on.
“Usually, I read what I’ve written to someone else before filing, but I was home alone on that fateful day, and in a hurry. So when I’d finished, I just pressed Send. And then, when the column appeared the next day, the landmine exploded.”
Mr Clarkson described the moment he picked up a copy of The Sun to “see what all the fuss was about”.
“We’ve all been there, I guess. In that precise moment when we suddenly realise we’ve completely messed up. You are sweaty and cold at the same time. And your head pounds. And you feel sick. I couldn’t believe what I was reading. Had I really said that? It was horrible," he continued.
“I knew what had happened straight away. I’d been thinking of a scene in Games Of Thrones, but I’d forgotten to mention this. So it looked like I was actually calling for revolting violence to rain down on Meghan’s head.
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“I was very angry with myself because in all those controversial days on Top Gear, when I was accused of all sorts of things, it was very rarely sexism.
“I was mortified and so was everyone else. My phone went mad. Very close friends were furious. Even my own daughter took to Instagram to denounce me.”
Clarkson’s statement went on to say ITV and Amazon, which airs Who Wants To Be A Millionaire and The Grand Tour respectively, were “incandescent”.
“The Sun quickly apologised and I tried to explain myself. But still, there were calls for me to be sacked and charged with a hate crime," he said.
“More than 60 MPs demanded action to be taken. ITV, who make Who Wants to be a Millionaire, and Amazon, who make the Farm Show and The Grand Tour, were incandescent.
“I therefore wrote to everyone who works with me saying how sorry I was and then on Christmas morning, I emailed Harry and Meghan in California to apologise to them too.
“I said I was baffled by what they had been saying on TV but that the language I’d used in my column was disgraceful and that I was profoundly sorry.”
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Clarkson, who referenced England football captain Harry Kane’s World Cup penalty miss, said he will “try” to be “interesting and vigilant” at the same time in future columns.
His statement concluded: “Over the last 30 years, I have written very nearly 5,000 newspaper and magazine columns, so it was inevitable that one day, I’d do a Harry Kane and sky one of the damn things. Which is what happened with the piece about Meghan.
“So can I move on now? Not sure. It’s hard to be interesting and vigilant at the same time. You never hear peals of laughter coming from a health and safety seminar. But I promise you this, I will try.
“Who knows? Very soon now I shall be a grandfather, so in future, maybe I’ll just write about that.”
The spokesman for Harry and Meghan said: “On December 25, 2022, Mr Clarkson wrote solely to Prince Harry, the Duke of Sussex. The contents of his correspondence were marked private and confidential.
“While a new public apology has been issued today by Mr Clarkson, what remains to be addressed is his long-standing pattern of writing articles that spread hate rhetoric, dangerous conspiracy theories and misogyny.
“Unless each of his other pieces were also written ‘in a hurry’, as he states, it is clear that this is not an isolated incident shared in haste, but rather a series of articles shared in hate.”
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Alongside hosting The Grand Tour, the TV presenter also stars in Amazon Prime TV series Clarkson’s Farm which documents his efforts to run the Oxfordshire Farm he bought in 2008.
The success of the series has seen visitors flock to the Diddly Squat farm shop in Chipping Norton Road, Chadlington, and the second series is due to launch on February 10 on Prime Video in more than 240 countries and territories.
However, Variety have reported that Amazon Prime Video will not be working with Clarkson beyond seasons of The Grand Tour and Clarkson’s Farm they have previously commissioned.
In 2018, Clarkson also took over from Chris Tarrant as the presenter of long-running ITV gameshow Who Wants To Be A Millionare?
The PA news agency understands ITV is due to film a new series of the show, which was agreed prior to Clarkson’s comments, but has no further commissioning commitments beyond that.
Speaking at a Broadcasting Press Guild event in December, managing director of ITV Studios Kevin Lygo said Clarkson’s comments about the Duchess of Sussex were “awful” but added the broadcaster plans to keep him as the host of the gameshow “at the moment”.
Amazon have been approached for comment.