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.