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.
After the tabloid published a story about his extramarital affair last month, Bezos ordered a team of private investigators to get to find out how the Enquirer obtained lurid texts between him and former TV anchor Lauren Sanchez.
In a post on blogging platform Medium Thursday, Bezos said Enquirer publisher American Media Inc (AMI), led by David Pecker, approached him with a threat to publish the photos if he did not halt an investigation into the motives behind that leak.
Bezos declined to do so.
"They said they had more of my text messages and photos that they would publish if we didn't stop our investigation", Bezos wrote in his blog post on Medium.com.
Mr. Bezos wrote that this week, the tabloid's editor, Dylan Howard, e-mailed an attorney for Mr. Bezos's long-time security consultant to describe photos the Enquirer "obtained during our newsgathering". Howard ends his email by saying, "It would give no editor pleasure to send this email. And if you don't do X quickly, I will report the embarrassing information".
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. "If in my position I can't stand up to this kind of extortion, how many people can?" he said.
The blog post is the latest twist in a weeks-long saga that has brought the personal life of Bezos, the richest person in the world, into the spotlight.
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However, Bezos said that "rather than capitulate to extortion and blackmail, I've chose to publish exactly what they sent me, despite the personal cost and embarrassment they threaten".
In its January 9 story, the Enquirer reported that Bezos sent "sleazy text messages and gushing love notes" to Sanchez, months before Bezos announced he was splitting up with his wife, MacKenzie.
On Feb. 5, The Post reported that Bezos and de Becker suspected that the source of the text and photo leaks may have been Sanchez's brother, Michael, a California public relations executive who is close to Pecker and various figures in Trump's orbit, including former campaign advisers Roger Stone and Carter Page. This, of course, refers to the $150,000 AMI reportedly paid Karen McDougal to cover up the story of her alleged affair with one Donald J. Trump. The company has been cooperating with investigators in the Southern District of NY ever since it admitted to making a $150,000 "catch-and-kill" payment to rid of a story about an alleged affair Trump had with former Playboy model Karen McDougal.
In September, the Justice Department agreed to a non-prosecution agreement with AMI, which requires the company and some top executives, including Pecker and Howard, to co-operate with authorities.
AMI has not yet responded to the BBC's request for comment.
In his post on Medium, Bezos called his ownership of The Post "a complexifier" for him.
Hours later, the first Enquirer story on his relationship with Sanchez posted online.
Sounds like soon some people at The National Enquirer are going to "learn to code".