NASHUA, N.H.— Robert F. Kennedy Jr. stood in front of a statue of President John F. Kennedy at the site marking his uncle’s first stop in the 1960 campaign. 

Then he defended his claim that a contaminant in water might be causing people to identify as transgender. In an interview later, he expounded upon his position. “If we’re chemically damaging sexual development in our children, if it injures frogs that way, my idea would be it should be prohibited unless they…show it doesn’t hurt people,” he said Tuesday. Kennedy was citing a small study in which frogs were exposed to a herbicide and their genders changed.

The maverick presidential run by the 69-year-old environmental lawyer for the Democratic presidential nomination is a mix of the nostalgic and the weird. 

While tapping into his family legacy as the scion of the most-storied family in Democratic politics, he is taking to the campaign trail to push viewpoints somewhere between outside the American mainstream and firmly in conspiracy-theory land.

‘I believe that my viewpoints are absolutely consistent with President Kennedy,’ says Robert F. Kennedy Jr.

He has aired claims—debunked by public-health experts—linking childhood vaccinations to autism. He has cast doubt on the safety of Covid-19 vaccines. He has questioned whether prescription drugs have caused a rise in school shootings and whether Wi-Fi exposure leads to cancer. He has said that the Central Intelligence Agency assassinated his uncle and could have killed his father, Robert F. Kennedy, despite no concrete evidence. And he has said the U.S. is perpetuating the Ukraine war to fuel the defense industry.

It isn’t likely that any of this will endear him to enough Democratic voters to replace Joe Biden as the party’s presidential nominee in 2024. But he is causing the Biden team, and the party, some heartburn as he campaigns in one of the earliest-voting states.

“I believe that my viewpoints are absolutely consistent with President Kennedy,” he said during an interview later Tuesday inside the New Hampshire State House. Kennedy said it’s not that his views are unconventional; it’s that the Democratic Party has changed. “I see myself as a mainstream liberal,” he said.

“Why are liberals now in support of war? Why are they suddenly in support of censorship? Why are liberals suddenly putting faith, ultimate faith, in pharmaceutical companies, who have always been great villains in the liberal zeitgeist? Why are liberals now aligned against medical freedom and bodily autonomy?” he asked.

On immigration and guns, Kennedy has offered solutions that put him more in line with Republicans than Democrats. He said he wouldn’t support taking anyone’s guns and would only support gun-control measures with broad bipartisan consensus. On immigration, he said that he would shut down the U.S.-Mexico border to stop a surge of illegal immigration and that he supports an expansive legal-immigration system.

Although Biden faces a lack of enthusiasm with Democratic voters, polling shows they remain aligned with him on policy. Democratic voters overwhelmingly trust Covid vaccines and support Biden’s handling of Ukraine. They back broad gun-control measures and favor less-restrictive immigration policies. 

Kennedy’s campaign sells stickers that say “I’M A KENNEDY DEMOCRAT,” and Tuesday night he gave a speech on foreign policy around the 60th anniversary of President Kennedy’s “Peace Speech.” He tied his foreign-policy platform—which includes ending involvement in Ukraine as fast as possible—to his uncle’s efforts to defuse Cold War tensions.

Robert F. Kennedy Jr. at the New Hampshire State House in Concord for a meeting with state legislators.

Robert F. Kennedy Jr. receiving a tour of the State House in one of the earliest-voting states.

Kennedy told reporters that an interview in which he aired vaccine skepticism and suggested that pollutants could cause people to identify as transgender was banned from YouTube. A YouTube spokesperson said portions of the interview, with the conservative Jordan Peterson, violated the website’s vaccine-misinformation policy. Kennedy’s Instagram account was removed over his Covid-19 vaccine views; it was reinstated once he announced his presidential bid.

Kennedy told The Wall Street Journal his views on vaccines have been misconstrued and said he is not antivaccine. He said he has all the required childhood vaccines but isn’t vaccinated against coronavirus and believes that vaccines should undergo more rigorous scrutiny. 

Biden is the clear favorite for the nomination. Polls have Kennedy in the high teens. The self-help author Marianne Williamson, another Democratic challenger, polls in the single-digits.

The Biden campaign and the Democratic National Committee declined to comment on Kennedy’s candidacy. The DNC has previously said it would support Biden and won’t sponsor any debates.

New Hampshire, among the first states in the Democratic nominating process, is where Kennedy has the best shot at beating Biden—if he has any shot at all. The state allows independents to vote in either party’s primary. 

Kennedy wore a suit with a skinny tie and Chelsea-style boots during his New Hampshire visit, and spoke in a measured style honed by years in the public eye. He said he is avoiding some types of campaign events including parades because his security team has determined that they aren’t safe given the risk of an assassination attempt by the CIA.

The hundreds of supporters who came to hear Kennedy’s foreign-policy speech—some who drove hours and came with copies of his books—spanned the ideological spectrum. 

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“All the things that he’s saying about bringing the country together, he really believes he really can do this, and he’s unlike anybody out there,” said independent voter Rebecca Giles, 54, a retired physician from Bedford, N.H., and a Kennedy campaign volunteer. Giles supported Donald Trump in the 2016 and 2020 New Hampshire primaries and general elections, but soured on him over his pandemic response, which she saw as heavy-handed. 

Michelle Roseto, 71, an independent voter from Milford, N.H., said she wants Democrats to put forward a stronger candidate than Biden but can’t support Kennedy. “I think his viewpoints, especially the antivax, which is the thing you really hear about, are extreme,” she said.

Robert F. Kennedy Jr., campaigning in New Hampshire, says he sees himself as a mainstream liberal.

On Tuesday, Kennedy met with New Hampshire Republican state lawmakers and was scheduled to speak to two libertarian groups. The New Hampshire Democratic Party recently sent a letter to Kennedy expressing “serious disappointment and grave concerns” over his decision to speak to a group working to limit government in the state. 

He is set to speak at a summit for Moms for Liberty, a group of conservatives advocating for greater parental involvement in schools.

“He’s running in the wrong primary,” Steve Bannon, a longtime Trump ally, said recently.

Write to Eliza Collins at eliza.collins@wsj.com