In a live televised forum billed as a conversation about the personal pressures of the presidency, a discussion about resilience and family veered into a tense exchange between former President Barack Obama and President Donald J. Trump over the role of marriage in public life.

The event, staged before a studio audience and moderated by a veteran journalist, was designed to offer a rare joint appearance: two presidents reflecting on leadership, sacrifice and the toll of the office. For much of the evening, the contrast between the men was stylistic rather than confrontational. Mr. Trump leaned forward in his chair, animated and expansive, describing the presidency as a test of stamina and strength. Mr. Obama, seated more loosely, responded in measured tones, often pausing before speaking.

When the moderator asked about resilience — what sustains a president during moments of crisis — Mr. Trump pivoted to a theme he has often invoked: family as proof of character.

“You can’t do this job without a great family,” he said. “Melania has been incredible. We have a strong marriage, a true partnership. That’s what gives you stability to lead.”

The comment was consistent with Mr. Trump’s longstanding practice of presenting his current marriage as evidence of personal steadiness. He has been married three times and has frequently described his present relationship in superlative terms, particularly in response to critics who question his temperament.

When the moderator turned to Mr. Obama and asked how important his partnership with former First Lady Michelle Obama had been during his time in office, he answered without hesitation.

“It was everything,” Mr. Obama said. “The White House can be a lonely place. A real partner makes the difference between performing leadership and living it.”

Then, after a brief pause, he added a note that subtly shifted the tone of the evening.

“That’s why,” he said, looking across the stage, “a president should be careful about using marriage as a credential.”

The audience stirred. Mr. Trump’s expression tightened.

Mr. Obama continued, framing his remarks not as gossip but as principle. “I’ve always believed private life deserves privacy,” he said. “But when you repeatedly present your relationship as a public badge of integrity, you invite the public to ask one simple question: Is the badge real?”

The moderator attempted to steer the conversation back to broader themes, but Mr. Trump interjected.

“Don’t talk about my wife,” he said sharply. “What are you doing?”

“I’m talking about accountability,” Mr. Obama replied evenly. “And the cost of image.”

In careful, abstract terms, Mr. Obama described what he called the tension between branding and transparency in modern politics. When personal relationships are used to reinforce political narratives, he suggested, they become part of the public story — and therefore subject to scrutiny.

“If taxpayers are funding a narrative,” he said, “they deserve transparency about what they’re funding.”

Mr. Trump rose slightly in his seat, visibly agitated. “Lies,” he said. “Disgusting lies. You have no right. My marriage is perfect. Perfect.”

He did not cite specific allegations, nor did Mr. Obama reference any particular claim. The exchange remained largely at the level of principle. Yet the confrontation felt personal.

Throughout his political career, Mr. Trump has tightly managed the public presentation of his family, often highlighting appearances alongside his wife and children at rallies and official events. At the same time, media coverage has occasionally noted separate schedules and extended periods during which the first lady remained out of the public eye, fueling speculation that the White House has generally declined to address directly.

Mr. Obama, by contrast, has typically avoided commenting on his successor’s personal life, adhering to a longstanding norm among former presidents to refrain from direct personal criticism. His decision to broach the subject, even obliquely, marked a departure from that restraint.

“You’re the one who made it public,” Mr. Obama said in closing the exchange. “You made it part of the pitch. And when you sell something as proof of fitness, people will ask to see what’s inside the box.”

For a moment, Mr. Trump appeared at a loss for a response. Then, in a gesture that startled the audience, he removed his lapel microphone and dropped it to the floor.

“This is a disgrace,” he said, before stepping away from the chair.

The broadcast cut to a wide shot as the moderator attempted to restore order. Producers later described the interruption as unscripted.

Political historians noted that while policy disputes between former presidents are common, public arguments over personal matters are rare. “There’s an unwritten rule of decorum,” said Barbara Perry, a presidential scholar at the University of Virginia. “Presidents may criticize each other’s decisions, but they generally avoid attacking family.”

Supporters of Mr. Trump accused Mr. Obama of crossing that line. Allies of Mr. Obama argued that he had addressed a broader issue: the use of personal narrative as political capital.

Whether the exchange will have lasting political consequences remains uncertain. What is clear is that a forum intended to explore the burdens of leadership became, instead, a stark illustration of how intertwined personal image and public authority have become in contemporary politics.

In the carefully managed theater of the presidency, family can symbolize stability. But as the evening demonstrated, once that symbol is placed at the center of a political argument, it may no longer remain untouched by it.