The 48-year-old posted a picture of the couple together outside the former French consulate building in Reykjavík, where a 1986 Cold War summit was held between Ronald Reagan and Mikhail Gorbachev.
His caption read: "'This is the house ware M Gorbachev R Reagan did some epic. it's also where S and I GOT MARRIED!" S refers to 24-year-old Brett, whose nickname is Scottie.
Charlie added the hashtag: "RumorMilUhPede". His beloved only added to the suspicion that it was all a big joke with her comments.
When one of her Twitter followers talked about the nuptials, she responded: "Hahahahahahahahahahaha epic."
Another post showed the 'newlyweds' embracing in front of a fire extinguisher. Alongside it, Charlie had written: "It's like screaming "Theater, in a crowded Fire!!"get over me,or get yuse to"Us"...
The wedding Tweet was followed by another showing the Anger Management star getting a Jaw tattoo, with his 'wife' holding his hand. The etching was the work of renowned tattoo artist Gary James.
He wrote: "What do u get wen u organize a Greg James a Sheenius and a Skottie so Hottie? J Ballard snapin a JAWS tat foto!" Jeff Ballard is the actor's publicist.
Charlie, a father of five, has been seeing Brett since November. He has two girls by his second wife Denise Richards, Sam, nine, and Lola, eight.
Then there are his twin boys, four-year-old Bob and Max, whose mother is his troubled third wife Brooke Mueller. They split up in 2011.
Meanwhile, his daughter by high school sweetheart Paula Profit is 30. Before Brett he had a number of live-in girlfriends, whom he referred to as goddesses.