Flanked by two magistrates, Judge Connell said: “If such a comment was made at a volume Lyons could hear, at that distance, with the music playing and people talking, no-one on the CCTV can be seen to react.
“We do not accept Mr Lyons’ evidence that a racial comment was uttered.”
The judge said that the footage showed Lyons “took steps to put himself into Mr Mackay’s personal space”, while the victim was looking at his mobile phone until the moment of the attack.
The evidence, it was ruled, contradicted Lyons’ accusation that Mr Mackay “walked up to me and mumbled” the antisemitic remark.
When he was initially convicted, Lyons received a 12-month restraining order. He was also fined £3,000 and ordered to pay £5,956 in legal costs.
After his failed appeal, he was ordered to pay an extra £3,000 fine and another £7,000 in legal costs.
Lyons, the chief executive of Enstar Capital Ltd, and the victim had been close friends for several years, until relations broke down in the summer of 2017.
During Tuesday's appeal hearing, new evidence was heard from Franklin Price, who had read about the case in the Daily Mail. He testified that he had heard Mr Mackay describe a Jewish lawyer a "kike" in 2015.
Mr Mackay denied holding antisemitic views, saying he had been engaged to a Jewish woman, and had worked on a kibbutz in his youth.
Jonathan Goldberg QC, who defended Lyons, called Mr Mackay an “arrogant, headstrong millionaire” with a “hedonistic lifestyle of Roman Emperor proportions”.
But prosecutor Irshad Sheikh accused the defence counsel of attempting to “blacken” Mr Mackay’s character, arguing: “These scurrilous allegations have no evidential basis”.
Mr Sheikh added: “[Lyons] has all the motive in the world to prejudice your minds against Mr Mackay. This case is a perfect example of kite-flying and innuendo.”
The court also heard that, in a police interview, Lyons admitted to feeling regret over slapping Mr Mackay “without thinking straight”, saying it was “quite silly”.
Mr Mackay is the founder of Hometrack, a property market analytics company, which was sold to Zoopla for £120million.
After the verdict, Lyons' solicitor Phil Smith said: “The use of physical force was an isolated incident and totally out of character. He accepts that it was wrong but he genuinely felt provoked and vulnerable.
"Regrettably, he slapped Mackay with an open hand and now wants to put the incident behind him and get on with his life.”
