"We need to know who else is there besides Jeff Bezos," he added.

"This is a warning volley over the bow that they need to know — who is their Tim Cook to this brilliant Steve Jobs?" the senior associate dean at the Yale School of Management said on " Power Lunch ." Cook succeeded Jobs as CEO of Apple.

In a blog post published Thursday, Bezos claimed the National Enquirer threatened to post sexual pictures the billionaire texted to his mistress, including a "below the belt selfie." He accused the tabloid's publisher, American Media Inc., of blackmail and extortion.

The ongoing drama surrounding Amazon CEO Jeff Bezos could be "mildly" harmful to the company's brand and should be a warning to the board to have a succession plan in place, management expert Jeffrey Sonnenfeld told CNBC on Friday.

But that doesn't mean Sonnenfeld thinks Bezos should go. He said the CEO may have been "indiscreet and foolish," but added, "How we judge somebody in a leadership role is — how do they respond and, of course, what's the magnitude of what they did."

In this case, Bezos was a "victim," he said. "This is not a situation [of] a guy who is abusing company resources or pilfering or harassing employees."

Bezos also handled the controversy "effectively," added Sonnenfeld. "He's taken the weapon away. Instead of being held hostage by these people, he's pretty much obliterated the issue. We think a little bit less of Jeff, but we are impressed, I think, with his resilience."

Federal prosecutors are now reviewing the National Enquirer's handling of the story, a source familiar with the matter told NBC News. They want to determine whether the tabloid violated an immunity agreement its parent company struck last year in the investigation into Michael Cohen, President Donald Trump's former personal lawyer. That deal required the company to agree to "commit no crimes whatsoever."

Branding and marketing strategist Dean Crutchfield told CNBC that the CEO has put his company "in a really bad spot" and noted this could "drag on" for some time.

"The richest man in the world could become the biggest joke in the world, and he certainly knows that," the CEO of Crutchfield and Partners told "Power Lunch." "So, there is panic running through his life and through his team."

The controversy isn't worrying Wall Street analysts. They don't think it will affect his ability to lead the company.

Shareholders, on the other hand, were a little jittery. The stock closed down 1.62 percent on Friday, adding to losses that began after Bezos announced on Jan. 9 that he and his wife of 25 years, MacKenzie, are divorcing. After that announcement, on Jan. 31, the online retailer reported mixed fourth-quarter earnings and issued weak guidance.

— CNBC's Tucker Higgins and Yun Li contributed to this report.