Bezos went on to say that, "Any personal embarrassment AMI could cause me takes a back seat because there's a much more important matter involved here", Bezos wrote. "I prefer to stand up, roll this log over, and see what crawls out".
A spokesman for the USA attorney's office in Manhattan did not immediately comment on Bezos' assertion that AMI had committed extortion.
Mr Bezos said AMI had wanted him to stop looking into it for political reasons.
The National Enquirer published a story last month that included lurid texts between Bezos and Sanchez.
The alleged emails show the outlet threatening to release a series of texts and photos he sent to his current girlfriend Lauren Sanchez while still married to MacKenzie Bezos. Pictures' "Argo" held at AMPAS Samuel Goldwyn Theater on October 4, 2012 in Beverly Hills, California.
In Mr Bezos' post on Thursday, he also published several e-mails between Mr de Becker's lawyer, Mr Martin Singer, and AMI's lawyer, Mr Jon Fine, and chief content officer, Mr Dylan Howard.
However, in the interests of expediating this situation, and with The Washington Post poised to publish unsubstantiated rumors of The National Enquirer's initial report, I wanted to describe to you the photos obtained during our newsgathering.
Among them, he claims: a bathroom selfie in which he's "wearing nothing but a white towel", "a full-length body selfie of Mr. Bezos wearing just a pair of tight black boxer-briefs or trunks", and a photo of Sanchez "smoking a cigar", possibly in a suggestive way.
"Rather than capitulate to extortion and blackmail", he wrote in a blog post on Medium, the executive published emails they sent proposing a "completely confidential" agreement. "It's a shocking and frightening thing Mr. Bezos has revealed".
Hours later, the first Enquirer story on his relationship with Sanchez posted online. The story carries the bylines of Howard and two reporters.
But Bezos was suspicious about how the tabloid could have possibly gotten access to his private exchanges.
Bezos is Amazon's largest shareholder, with a stake of about 16 percent. Or at least that's what the top people at the National Enquirer thought. The person wasn't authorized to discuss the matter publicly and spoke on condition of anonymity.
Sanchez, who is also his sister's manager, has declined to speak with The Associated Press on the record and did not immediately respond to an email seeking comment Thursday. "In a tweet, he said de Becker "spreads fake, unhinged conservative conspiracy theories" and "'dog whistle' smears". It admitted to "catch-and-kill" practices as part of a deal with federal prosecutors, who agreed to not pursue charges against the company.
According to the New York Times, Trump and Pecker dined together to celebrate him winning the presidency. The company then intentionally suppressed the story until after the 2016 election.
Since the tabloid affair, Bezos opened an investigation into the leak led by Gavin de Becker, a public safety expert and former appointee of US President Ronald Reagan. The deal not to prosecute was conditional on AMI not committing another crime, the office said.