Horne called Kuenssberg a "disingenuous plopcarpet" and told her to resign following her coverage of the General Election.
The actor, who is set to appear on BBC One in the hotly-anticipated Gavin And Stacey Christmas special, posted the tweets in the early hours of Friday as the election results came in. He has now deleted them.
On Sunday he wrote on Twitter: "I apologise to both @bbclaurak and the BBC for my comments on election night.
I apologise to both @bbclaurak and the BBC for my comments on election night. I was caught up in the frenzy of the election and realise I used my public platform and passion for politics in an incorrect way. I regret this & we should stand together. Merry Christmas one and all.— Mathew Horne (@mfhorne)December 15, 2019
"I was caught up in the frenzy of the election and realise I used my public platform and passion for politics in an incorrect way. I regret this & we should stand together. Merry Christmas one and all."
In a reply, Kuenssberg accepted Horne's apology.
She said "no-one wins when we scream and shout rather than talk - maybe Twitter could move on from being Bitter in 2020", adding that she was "looking forward" to the Gavin and Stacey festive episode.
It is reported that Kuenssberg was discussing the Labour party losses around the time of Horne's tweet.
Horne retweeted a message from the political editor in which she said that the "swing in the North of England is pretty consistently v big Labour to Tory", and wrote: "Let's see... your 'journalism' knows no bounds.
"Resign, you disingenuous plopcarpet."
Horne will reprise his role as Gavin Shipman in the Christmas special of the popular sitcom, which is set to air on BBC One on Christmas Day.