Mr Bezos said the magazine's parent company American Media Inc (AMI) wanted him to drop an investigation into how it obtained his private messages.
Mr Bezos and his wife, MacKenzie, said last month they were getting divorced.
Their announcement came just before the National Enquirer carried reports about the tech titan's extramarital affair.
AMI has not yet responded to the BBC's request for comment.
What does Bezos claim?
In a stunning blog post on Thursday, Mr Bezos posted an email he said had been sent to his intermediaries by AMI's representatives threatening to publish "intimate photos" of Mr Bezos and his lover, former TV host Lauren Sanchez.
The billionaire, who also owns the Washington Post, said AMI had wanted him to make a "false public statement" that the National Enquirer's coverage of him and his mistress was not politically motivated.
According to emails included by Mr Bezos in his blog, an AMI lawyer proposed on Wednesday that the photos would not be published in return for a public statement "affirming that [Bezos and his team] have no knowledge or basis" to suspect such a motive.
"Rather than capitulate to extortion and blackmail," wrote Mr Bezos, "I've decided to publish exactly what they sent me, despite the personal cost and embarrassment they threaten."
Early in the blog post, Mr Bezos mentions AMI's links to President Donald Trump.
Why does he mention President Trump?
Mr Bezos said his ownership of the Washington Post was a "complexifier" for him because he had made enemies of "certain powerful people", including President Donald Trump, who is a friend of AMI's boss David Pecker.
AMI recently admitted it had co-ordinated with the Trump presidential campaign to pay a Playboy model $150,000 (£115,000) in hush money to keep quiet about her alleged affair with Mr Trump.
Mr Bezos notes in his blog post how the publisher confessed to the so-called "catch and kill" deal to bury Karen McDougal's politically embarrassing story.
AMI's agreement to co-operate with federal authorities means it will not face criminal charges over the payments, Manhattan prosecutors announced in December.
Mr Trump's former lawyer Michael Cohen - who facilitated the hush money at the direction, he says, of Mr Trump - has already admitted violating campaign finance laws.
What about Bezos' reputation?
The Amazon boss didn't try to hide the potential for embarrassment, writing "of course I don't want personal photos published" and noting what he called "AMI's long-earned reputation for weaponising journalistic privileges".
"But," he continued, "I also won't participate in their well-known practice of blackmail, political favours, political attacks, and corruption. I prefer to stand up, roll this log over, and see what crawls out."
His blog contained itemised details of 10 pictures in an email from the magazine's editor Dylan Howard, who said they had been "obtained during our newsgathering".
New Yorker writer Ronan Farrow alleged that he "and at least one other prominent journalist" had been subject to similar threats from AMI.
Mr Bezos said "AMI's claim of newsworthiness is that the photos are necessary to show Amazon shareholders that my business judgment is terrible".
But the Amazon boss countered that the firm's results "speak for themselves".
Dylan Howard's name, along with two National Enquirer reporters, appears on a story the magazine published on 9 January containing alleged details of Mr Bezos' affair with Ms Sanchez.
