Robert F. Kennedy Jr. apologized for a leaked video of the phone call, in which Donald Trump questions the safety of childhood vaccines and says the injury to his ear “felt like the world’s largest mosquito.”

A leaked video of a phone call between former President Donald J. Trump and the independent presidential candidate Robert F. Kennedy Jr. on the day after the attempted assassination of Mr. Trump offered a behind-the-scenes look into the former president’s efforts to coax Mr. Kennedy out of the race and into his camp.

The video was first posted early on Tuesday morning by Mr. Kennedy’s eldest son, Bobby Kennedy III, and then swiftly deleted, according to screenshots of the original message and a person briefed on the matter. The younger Mr. Kennedy criticized Mr. Trump for his vice-presidential pick and said he wanted to expose his “real opinion” on vaccinations.

The video provides a glimpse of the private relationship between Mr. Trump and Mr. Kennedy, who have long been friendly, though Mr. Trump has publicly criticized Mr. Kennedy during the campaign.

Mr. Kennedy apologized for the leaked video almost immediately on X, the same social media site where the video first appeared on Tuesday morning.

“When President Trump called me I was taping with an in-house videographer,” he wrote. “I should have ordered the videographer to stop recording immediately. I am mortified that this was posted.”

Over Mr. Kennedy’s speaker phone, Mr. Trump, the Republican nominee, can first be heard describing scientifically baseless concerns about vaccinations — an issue Mr. Kennedy is closely associated with — expressing disbelief at the volume of vaccinations given to infants.

“I want to do small doses,” Mr. Trump said, before observing that the practice of giving multiple vaccinations at once means the shot “looks like it’s meant for a horse, not uh, you know, a 10-pound or 20-pound baby.”

Mr. Kennedy and the organization he co-founded, Children’s Health Defense, have for years promoted claims about the health risks of childhood vaccines that have repeatedly been refuted by studies.

“And then you see the baby all of a sudden starting to change radically, I’ve seen it too many times,” Mr. Trump said. “And then you hear that it doesn’t have an impact, right? But you and I talked about that a long time ago.”

Then Mr. Trump pivoted to an apparent effort to win his support: “I would love you to do something. And I think it’ll be so good for you and so big for you. And we’re going to win.”

Mr. Kennedy, standing while wearing a shirt and tie, says “Yeah,” barely audibly. Throughout the video, Mr. Kennedy says little else.

Mr. Trump also briefly described his recent call with President Biden and said he had asked about the near miss of the bullet in Pennsylvania. “It was very nice, actually,” Mr. Trump said. “He called me, and he said, ‘How did you choose to move to the right?’”

Mr. Trump marveled that he had been shot by an AR-15-type weapon, the sort of military-style rifle Mr. Biden wants to ban — “pretty tough guns, right?”

Of the wound to his ear, Mr. Trump said: “It felt like the world’s largest mosquito.”

In his original post on X, the younger Mr. Kennedy wrote: “I am a firm believer that these sorts of conversations should be had in public. Here’s Trump giving his real opinion to my dad about vaccinating kids.” Bobby Kennedy’s wife, Amaryllis Fox, is Mr. Kennedy’s campaign manager.

On Monday, Mr. Trump and Mr. Kennedy met in person in Milwaukee ahead of the Republican National Convention, which stirred rumors that Mr. Kennedy was planning to drop out of the race and endorse Mr. Trump.

Afterward, though, Mr. Kennedy said he had no plans to drop out.

Mr. Kennedy is in the thick of an expensive and time-consuming effort to get on the ballot in all 50 states. His persistence as a candidate has unnerved both Mr. Trump’s camp and Mr. Biden’s — recent polls suggest that Mr. Kennedy could pull votes equally from both candidates, including in swing states.