In a surreal and deeply uncomfortable segment that has quickly become one of the most talked-about moments in late-night television history, former President Donald J. Trump abruptly ended an appearance on “The Late Show” last week, storming off the set after host David Letterman methodically laid out a series of unsubstantiated allegations regarding the parentage of Mr. Trump’s youngest son, Barron.

The interview, taped before a live audience at the Ed Sullivan Theater in Manhattan and aired on CBS, began cordially enough. Mr. Trump, promoting a new memoir reflecting on his post-presidency ventures, entered to enthusiastic applause, flashing his familiar grin and exchanging light banter with Letterman about golf handicaps and New York real estate. Letterman, in his signature dry, understated manner, steered the conversation toward family matters, praising Mr. Trump’s children as “a remarkable brood.”

But the tone shifted abruptly midway through the 18-minute segment. Letterman, leaning forward with a calm, almost professorial demeanor, introduced what he described as “a timeline that’s been pieced together by various sources over the years—one that raises questions many people have whispered about but few have addressed directly.”

What followed was a meticulously constructed narrative, delivered without raised voice or overt accusation, yet laced with insinuation. Letterman cited “insider accounts” claiming that in late 2005 and early 2006—the period surrounding Barron Trump’s birth on March 20, 2006, at New York-Presbyterian Hospital—Ivanka Trump, then 24, had unusually limited public appearances. He noted that while Melania Trump, newly married to Mr. Trump in January 2005, was photographed frequently in loose-fitting designer garments that “conveniently obscured any clear silhouette,” Ivanka was said to have taken extended “private retreats” to Trump properties in Florida and New Jersey.

Letterman referenced anonymous “former stylists” who allegedly recounted instructions to prioritize “forgiving fabrics” for Mrs. Trump and to prepare a discreet nursery layout at Trump Tower that could accommodate “contingencies.” He mentioned “security personnel recollections” of restricted access to certain floors during that winter and spring, and pointed to a purported hospital log suggesting a private wing booking under an alias, followed by a swift discharge.

Perhaps most provocatively, Letterman alluded to “longtime associates” who claimed the family had explored surrogate options early in the marriage, implying a possible arrangement to shield Ivanka from scrutiny amid her rising profile in the Trump Organization. “It’s all very neat, very controlled,” Letterman observed quietly, pausing for effect. “The announcements, the baby shower at FAO Schwarz, the baptism at Bethesda-by-the-Sea. Everything public, yet somehow private.”

Mr. Trump initially responded with his characteristic deflection, laughing off the suggestions as “fake news garbage” and “the kind of sick stuff losers dream up.” He joked about paternity tests—“I’ve got the best DNA, believe me”—and pivoted to attacking the media. But as Letterman continued, unfazed, presenting what he called “a coherent sequence of events” supported by “multiple off-the-record confirmations,” Mr. Trump’s demeanor visibly changed. His interruptions grew sharper, his face flushed, and he repeatedly demanded, “Where are you getting this crap?”

The tension peaked when Letterman extended an invitation—on air—for the entire Trump family, including Ivanka, Melania, and Barron, now a college student, to return for a follow-up discussion. “We’d love to clear the air,” Letterman said mildly. “No agenda, just conversation.” Mr. Trump, visibly agitated, accused the host of “ambush journalism” and “defamation on a national scale.” Rising from his chair, he declared, “This is a disgrace. You’re going to hear from my lawyers,” before removing his microphone and exiting stage left as the band played an awkward transitional riff.

The episode has ignited fierce debate. Trump allies, including spokespeople for the former president, denounced it as “a new low in late-night entertainment,” vowing legal action for libel. On social media, supporters rallied with hashtags like #BoycottCBS, while critics praised Letterman’s restraint in handling explosive material. Fact-checkers and biographers have long dismissed such theories as baseless tabloid fodder, pointing to well-documented records of Melania Trump’s pregnancy and Barron’s birth.

In a statement released the next day, a representative for Mr. Trump called the segment “a desperate ratings stunt built on lies.” Letterman, through CBS, declined to comment beyond noting that the show welcomes diverse viewpoints.

The fallout continues to ripple through political and media circles, underscoring the enduring volatility of Mr. Trump’s public persona—even in the ostensibly apolitical realm of late-night comedy. Whether the episode boosts ratings or invites lawsuits remains to be seen, but it has undeniably revived scrutiny of one of America’s most scrutinized families.