The comedy legend told how the man flashed a pistol at him and said he was going to get revenge after his pet disappeared.
Connolly, 78, said he was convinced the man, who lives along the street from him in the Florida Keys, was serious about carrying out the threat.
The retired stand-up also said the Donald Trump-supporting eccentric character was stockpiling guns at his home in preparation for a civil war.
He said: “I met a guy, he lives along the street from me, and I see him sometimes when he’s working on his house.
“His cat had disappeared and then I met him about two or three days later and I said ‘Did you find your cat?’.
“He said ‘No, but I know who’s got it,’ and he pulled a gun out of his pocket.
“It was breathtaking. He meant it, he was going to shoot the guy.
“He reckoned the guy had killed his cat for whatever mischief it had been up to and he was going to shoot him.”
“I met him later when it was election time and he was on Donald Trump’s side and he reckoned there was going to be a civil war.
“He told me he had seven guns.”
In an interview on the BBC Headliners podcast, Connolly said he had become friends with the man despite disagreeing with his views on the gun culture in America.
He added: “I like him, he’s a good neighbour. He likes to sit down and have a chat when you are passing.
“He works very hard on his house and his life’s work is keeping his house running.
“There’s big iguanas that come along and he likes to shoot them. He hates them and says they don’t belong there and they are not naturally part of the scene so he shoots them and throws them into the water.
“He is delighted he will tell you what the score is today and say ‘I got four’. He’s an interesting guy.”
Connolly released his autobiography Windswept and Interesting earlier this month.
He announced his retirement from stand-up in 2018 but has continued to make documentaries and has just filmed a series called Billy Connolly Does.. for TV channel Gold.
He was diagnosed with Parkinson’s disease nine years ago after a doctor spotted him walking strangely through the lobby of a hotel in Los Angeles.
He moved to Florida from the couple’s previous home in New York after doctors advised him to live in a warmer climate.
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