Clarkson wrote: “I hate her (Meghan). Not like I hate Nicola Sturgeon or Rose West. I hate her on a cellular level.”
He went on to say he was “dreaming of the day when she is made to parade naked through the streets of every town in Britain while the crowds chant, ‘Shame!’ and throw lumps of excrement at her”.
Speaking to broadcasters on Monday, Ms Sturgeon said his comments about Meghan were “beyond the pale”.
She said: “I accept I’m a politician and people get to say all sorts of things about me and that’s part of the democratic process.”
Mr Dolan continued: “But this guy is paid to have an opinion. And he’s paid to be outrageous. He’s one of the country’s best-known and most engaging journalists and TV presenters, and it’s not an accident why.”
However, Clarkson’s daughter, Emily, 28, joined a host of stars criticising the former Top Gear star.
She reshared online a previous post in which she revealed her “steadfast defence” of the Duke and Duchess of Sussex. Her comments also included criticism of the media for being “hellbent on destroying” Meghan.
On Sunday, she captioned the post: “I said it last week, I’ll say it again. I believe this on a cellular level.”
In a follow-up posted on Instagram Story, she continued: “My views are and have always been clear when it comes to misogyny, bullying and the treatment of women by the media.
“I want to make it very clear that I stand against everything that my dad wrote about Meghan Markle and I remain standing in support of those that are targeted with online hatred.”
Columnist Emma Woolf also condemned the Grand Tour host’s comments, claiming on GB News: “I think Jeremy Clarkson has made a complete fool of himself and where was his editor? Why was it allowed to go out?”
She added: “The silly thing is, Jeremy Clarkson is just feeding into the Harry and Meghan hysteria. The best thing he could have done was ignore them instead he’s feeding this fantasy they’ve got that they whole media is against them and that everybody is threatening them and treating them badly.
“Frankly, if they wanted privacy they should have got on with having privacy rather than signing all these deals.”
Clarkson’s comments also angered fans of Meghan and Harry when an excerpt was shared on Twitter.
A user of the platform called Hannah commented: “In what world is something so vicious and unhinged allowed to be printed? Completely vindicates everything in #HarryandMeghanNetflix.”
It was then shared by Omid Scobie, who is best known for co-authoring the Harry and Meghan biography Finding Freedom.
He tweeted: “That would be the same Jeremy Clarkson who was at a private Christmas party with Camilla, the Queen Consort earlier this week (alongside guests including a Daily Mail editor and Piers Morgan).”
Clarkson was among stars, including Dame Judi Dench, Strictly’s Claudia Winkleman and Dame Maggie Smith, who attended a Christmas lunch with the Queen Consort on Wednesday.