The newspaper, the couple said, had not contacted Meghan to say sorry and they said they thought the paper needed to change its coverage and ethics for there to be meaning behind the words.
In the column published last week, Clarkson, who gained worldwide fame as a presenter of motoring show Top Gear, wrote that he had dreamt of Meghan being paraded through British towns and publicly shamed. “Everyone who’s my age thinks the same way,” he wrote.
A spokesperson for Prince Harry and Meghan said in a statement on Saturday that The Sun exploited hate.
“While the public absolutely deserves the publication’s regrets for their dangerous comments, we wouldn’t be in this situation if The Sun did not continue to profit off of and exploit hate, violence and misogyny,” a spokesperson for Harry and Meghan said.
“A true apology would be a shift in their coverage and ethical standards for all. Unfortunately, we’re not holding our breath.”
The Duke and Duchess of Sussex stepped down from royal duties in March 2020, saying they wanted to make new lives in the United States away from media harassment.
In a Netflix documentary series, Meghan spoke about how her treatment by the media had left her feeling suicidal, as well as concern over whether she and her children were safe.
More than 60 lawmakers signed a letter written by Caroline Nokes, chair of parliament's Women and Equalities Select Committee, to the editor of The Sun, warning such articles contribute to a climate of hatred and violence against women.
Britain's Independent Press Standards Organisation regulator said on Tuesday that it had received more than 20,800 complaints, the most about any article since it was established in 2014.
Clarkson has separately said he was “horrified to have caused so much hurt” following the backlash and that he would “be more careful in future”.
On Friday, the newspaper issued a statement: “In last Saturday's Sun, Jeremy Clarkson wrote a comment article about the Duchess of Sussex. It provoked a strong response and led to a large number of complaints to Ipso, the independent press regulator.
“In a tweet earlier this week, Jeremy said he had made a “clumsy reference to a scene in Game of Thrones”, which had “gone down badly with a great many people” and he was “horrified to have caused so much hurt”. He also said he will be more careful in future.
“Columnists' opinions are their own, but as a publisher we realise that with free expression comes responsibility. We at The Sun regret the publication of this article and we are sincerely sorry.”
The newspaper added that the article had been removed from its archives and website.