Jeff Bezos, founder of Amazon.com Inc., accused National Enquirer’s publisher of trying to blackmail him by threatening to release embarrassing photos, escalating a fight that began with the tabloid’s revelations of his alleged extramarital affair.

In a lengthy and highly personal post on the web platform Medium, Mr. Bezos alleges that American Media Inc. demanded that he call off investigators he brought in to determine how the Enquirer obtained his personal text messages for the initial article it published and whether...