At one point, Clements said: "Sit on it, Chachi" - a reference to Baio's 1980s Happy Days character.
He responded with: "That's all you've got Kiwi? Bet you live with a lot of sheep."
Baio also shared a close-up photo of Clement and said: "It's nice to be thin & have all of my hair at 56. WTF happened to you? Born in 1974. Yeah OK."
He replied: "Bless you, Chachi. I hope your long battle with puberty is over by the time you hit 60."
He also shared a picture of Baio posing with Trump, alongside a picture of Adolf Hitler with leader of the Italian Fascist Party Benito Mussolini and said: "Were you guys going for this? "Let's do a Germans and Italians back together again thing!"
Baio responded with: "I stand with Israel. As does President @realDonaldTrump. Keep sucking the tit of @CNN. Worry about your country."
Baio has supported Trump's campaign throughout.
He claims he was attacked on his way to the Liberty Ball following Trump's inauguration on January 20.
A protester reportedly labelled him a 'f***ing fascist', and lunged at him.
Baio last year spoke at the opening night of the Republican National Convention.
In an interview on Fox last year, Baio described President Barack Obama as being "either dumb, a Muslim, or a Muslim sympathiser, and I don't think he's dumb".
He questioned what the "liberal endgame is" and added: "Is it to totally eliminate the United States as it was created and founded and the way it is now?"
In December, he accused Nancy Mack, the wife of Chad Smith, drummer for Red Hot Chili Peppers, of assaulting him at their children's school function after she cursed him over supporting Trump.
Mack has said she was trying to show him how Trump hugs women and has denied any physical aggression.
