In a statement on Monday, he said he emailed the couple on Christmas Day to say his language had been "disgraceful" and he was "profoundly sorry".
In the column, he wrote about a naked Meghan being pelted with excrement.
A spokesperson for Harry and Meghan said the article was not an isolated incident for Clarkson.
Clarkson had released a statement before Christmas saying he "was horrified to have caused so much hurt".
The Sun newspaper has also apologised for the December article and removed it from its website.
In the column, Clarkson wrote that he lay in bed "dreaming of the day when she [Meghan] 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".
He also claimed "everyone who's my age thinks the same way", and that her appeal to young people who "think she was a prisoner of Buckingham Palace" made him "despair".
A record 25,000 complaints have been made to press regulator Ipso since the piece was published.
In his lengthy Instagram post on Monday, the presenter of Amazon Prime's The Grand Tour and Clarkson's Farm and ITV's Who Wants to be a Millionaire?, said he usually reads what he's written before filing his copy, but he was home alone that day and in a hurry.
"So when I'd finished, I just pressed send. And then, when the column appeared the next day, the land mine exploded."
He said he picked up a copy of The Sun and quickly realised he had "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 said he had been thinking of a scene in Games of Thrones when he wrote about imagining the duchess being abused in the street, but had forgotten to mention it.
"So it looked like I was actually calling for revolting violence to rain down on Meghan's head."