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.
Bezos declined to do so.
According to the letters, in exchange for not publishing these photos, AMI's Deputy General Counsel, Jon Fine, wanted Bezos and his team to acknowledge that "they have no knowledge or basis for suggesting that AMI's coverage was politically motivated or influenced by political forces, and an agreement that they will cease referring to such a possibility".
Bezos wrote that this week, the tabloid's editor, Dylan Howard, emailed an attorney for Bezos' longtime security consultant to describe photos the Enquirer "obtained during our newsgathering".
It would give no editor pleasure to send this email.
"If in my position I can't stand up to this kind of extortion", he wrote, "how many people can?"
The post, which was linked from Bezos' Twitter account, was a remarkable escalation of a spat between the world's richest man and a tabloid known for its loyalty to the president of the United States.
"Rather than capitulate to extortion and blackmail, I've made a decision to publish exactly what they sent me, despite the personal cost and embarrassment they threaten".
His personal investigators, led by his longtime security consultant, Gavin de Becker, concluded that Bezos' phone wasn't hacked.
In recent interviews, including with The Daily Beast and The Washington Post, Mr de Becker has said he was investigating whether Ms Sanchez's brother, who has said he supports Mr Trump, may have been behind the leak for political reasons. 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.
AMI did not immediately respond to a request for comment.
"I and at least one other prominent journalist involved in breaking stories about the National Enquirer's arrangement with Trump fielded similar "stop digging or we'll ruin you" blackmail efforts from AMI", Farrow tweeted Thursday evening.
According to the New York Times, Trump and Pecker dined together to celebrate him winning the presidency.
Pecker is now cooperating with federal investigators who are looking into the company's involvement with the Trump campaign. The deal not to prosecute was conditional on AMI not committing another crime, the office said.
· Mr Bezos face selfie at what appears to be a business meeting.
In its January 9 story, the Enquirer said reporters followed Bezos and Sanchez "across five states and 40,000 miles" and "tailed them in private jets, swanky limos, helicopter rides, romantic hikes, five-star hotel hideaways, intimate dinner dates and "quality time" in hidden love nests".
"It shows how complicated and unsafe it is to make an agreement with National Enquirer", Levenson said.