getty (2) Eric Adams and Rudy Giuliani
While Giuliani, 78, described the incident as akin to being "shot," video surveillance of the incident shared by outlets including the New York Post shows the man approaching Giuliani from behind and slapping him on the back.
Giuliani told The Curtis Sliwa Show of the alleged assault: "I went forward, but luckily I didn't fall down," adding that if he were not in "pretty good shape," he would have "hit the ground and probably cracked my skull."
He claimed in a Facebook Live Monday morning that the slap "hurt tremendously" and that he did not know what it was. "I got hit on the back as if a boulder hit me," Giuliani said.
Giuliani said in his Live that the man — who has since been identified by police as 39-year-old Daniel Gill, according to ABC 7 NY — called him "dirty curse words" and was "menacing."
Giuliani continued: "So I said, 'Let's get him arrested. Let's make an example out of him.' ... This has to stop, this idea that you attack people."
Police confirmed to PEOPLE earlier this week that Gill had been arrested following the incident, and was charged with second-degree assault. The New York Times reports that prosecutors have since downgraded the charges to third-degree assault, second-degree harassment and third-degree menacing.
Speaking to reporters on Tuesday, Adams said: "I looked at the video, and someone needs to remind former Mayor Giuliani that falsely reporting a crime is a crime."
He elaborated on those remarks in an appearance on The Daily Show with Trevor Noah, saying the video showed that "it was not" assault.
"We are joking about it, but think about this for a moment," Adams added, comparing the incident to that of Amy Cooper, a woman who called the police on a Black bird watcher in Central Park in 2020 and was ultimately charged with filing a false police report.
"[Giuliani] falsely reported a crime and the district attorney should take that seriously. That person that he falsely reported spent 24 hours in jail. That's not acceptable."
Adams added that he would be calling the district attorney to look into the matter.
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"We must be consistent," he said. "All the theatrics that he did — that's not acceptable. If that tape wasn't there, imagine what would have happened to that man."
Police told CBS2 that Giuliani was at the supermarket to campaign for his son Andrew Giuliani, a New York governor hopeful who lost the GOP primary race on Tuesday.