Moore appeared the third episode in the series that debuted on July 15, which aired on Showtime. In the show, Cohen assumes different alter egos to interview political figures -- in Moore's case, acting as Gen. Erran Morad, an "anti-terror expert" and former agent of Israeli's intelligence agency, Mossad.
The show begins as Cohen, in full transformative makeup, discusses the importance of technology in the military.
"Modern technology is at the forefront of the fight against the war," he says in a fake Israeli accent. "One time, I killed a suicide bomber with an iPad 4."
Once Cohen sits down with Moore, he begins to tell the former Chief Justice of the Supreme Court of Alabama about a new technology that Israel has developed to "identify other abnormalities."
Cohen tells Moore that "sex offenders, and particularly, pedophiles, secrete an enzyme" that is about "three times the level of non-perverts," adding that authorities in Israel have developed a "machine" that is now used in schools and playgrounds that detects the when the enzyme is present.
"And if they detect a pedophile, the wand alerts the law enforcement and schools within a 100-mile radius."
Cohen then pulls out a wand that resembles the handheld metal detectors often used by security guards, scanning himself before scanning Moore, which then appears to cause the device to beep.
Cohen feigns confusion as the machine beeps for a second and third time after nearing Moore, asking him if he had happened to lend his jacket to someone else.
"Maybe," Moore responds. "I've been married for 33 [years] and never had an accusation for such things."
Cohen tells Moore that he's not accusing him "at all," prompting Moore to say, "Certainly, I'm not a pedophile, OK?"
Moore then says, "Maybe Israeli technology hasn't been developed properly," shaking Cohen's hand and ending the interview.
The video has been viewed on YouTube more than 3.8 million times since it was published on July 29.
According to the lawsuit, Cohen, "while in character, falsely and fraudulently induces unsuspecting victims, such as Judge Moore to be interviewed under dishonest, unethical, illegal and false pretenses."
In addition, the lawsuit, filed in the U.S. District Court in Washington, D.C. on Wednesday, alleges intentional infliction of emotional distress and fraud by the defendant and that Cohen sets the "unsuspecting victims up for ridicule" to promote his work and "to generate large financial returns for himself."
Moore's wife, Kayla Moore, is also listed as a plaintiff in the lawsuit.
Cohen "fraudulently" convinced Moore and his wife to travel to Washington, D.C., to film the show on Feb. 14 by telling them that Moore was to receive an award for "his strong support of Israel in commemoration of its 70th anniversary as a nation state," the court documents state.
Cohen also told the couple that the piece would appear on "Yerushalayim TV," a network that doesn't exist, according to the lawsuit.
Moore would not have agreed to the interview had he known Cohen's true identity and that it was a set-up, the complaint states.
After Moore discovered the truth, his attorney sent warnings to Cohen, Showtime and CBS that he would "resort to appropriate legal remedies if they chose to air the segment," according to the complaint.
The lawsuit contends that Cohen "falsely painted, portrayed, mocked and with malice defamed Judge Moore as a sex offender, which he is not." The couple filed the lawsuit because "they have been directly affected by the unlawful conduct" stated in the complaint, according to the court documents.
Moore has been the subject of "widespread ridicule and humiliation and has suffered severe loss of reputation" as a result of the show, according to the court documents.
Moore is suing for damages of $95 million and has asked for as a jury trial.
Showtime has not been served with the complaint, a spokesperson told ABC News, declining to comment further on the pending litigation.
Representatives for Cohen and CBS did not immediately respond to ABC News' request for comment. CBS is the parent company of Showtime.
Former Alaska Gov. Sarah Palin also appeared on the show, slamming the prank in which Cohen appeared as a disabled U.S. veteran in a wheelchair as "evil" and "sick."
Moore ran for the U.S. Senate as a Republican in 2017, despite sexual misconduct allegations by at least eight women, some of whom claim they were in their teens -- and Moore in his 30s -- when the misconduct occurred. Moore denied the allegations throughout the race.