The foulmouthed refusal to apologize to first lady Melania Trump by the erstwhile first son, Hunter Biden, sets the stage for an escalating legal battle between the high-profile presidential relatives.

“F— that! That’s not going to happen,” is what Mr. Biden told a British journalist, Andrew Callaghan, with respect to an apology. Mr. Callaghan was Mr. Biden’s interlocutor in July when, in comments captured on YouTube, the son of the 46th president suggested that the disgraced financier Jeffrey Epstein introduced Mrs. Trump to her husband, now America’s 47th president. 

Mr. Biden has offered no substantiation for his assertion — only citing remarks by the journalist Michael Wolff on a Daily Beast podcast. Meanwhile, Mr. Trump has called Mr. Wolff “mentally deranged” and “a total loser.” Mr. Wolff claims to have interviewed Epstein extensively before his arrest and death.

Mr. Biden’s remarks about the first lady prompted a lawyer for Mrs. Trump, Alejandro Brito, to pen a letter to Mr. Biden in which he called the accusations “extremely salacious.” Mr. Brito alleges that the speculation over how the president and the first lady were introduced caused the first lady “to suffer overwhelming financial and reputational harm.” The lawyer also accuses Mr. Biden of a “vast history of trading on the names of others,” an allusion to his famous father.

Mr. Brito warns that Mr. Biden, who secured an expansive pardon from his father that saved him from prison, could be liable to Mrs. Trump to the tune of more than $1 billion in damages for the declaration that “Epstein introduced Melania to Trump. The connections are, like, so wide and deep.” The first couple claim to have been introduced by a modeling agent, Paolo Zampolli, at a New York Fashion Week party in 1998. 

Now Mr. Biden explains that “what I said was what I have heard and seen reported and written, primarily from Michael Wolff but also dating back all the way to 2019 when The New York Times — I think Annie Karni and and Maggie Haberman — reported that sources said that Jeffrey Epstein claimed to be the person to introduce Donald Trump to Melania at that time.” That article, as Mr. Biden notes, sources to Epstein the claim that he “introduced Mr. Trump to his third wife, Melania.”

Mr. Biden adds that “the primary source was the interviews that Michael Wolff has been conducting, in which he has actually tapes of, I think, hours and hours and hours of interviews with Jeffrey Epstein directly. … I don’t think that these threats of a lawsuit add up to anything other than a designed distraction. … I don’t know how that in any way rises to the level of defamation to begin with.”

Mr. Biden, who has a law degree and has practiced at a white shoe firm, is likely alluding to the high bar public figures must clear if they are to succeed in defamation cases. The Supreme Court in New York Times Company v. Sullivan, a case from 1964, ruled that such a claim requires the showing of “actual malice.” That means that a defendant either knowingly made a false statement or made a claim with a “reckless disregard” for truth or falsity. The standard is meant to thwart spurious suits.

Mrs. Trump’s husband has of late succeeded in defamation cases — settling with ABC News for $15 million and CBS News for some $16 million — though the fate of those cases could owe more to the White House’s clout than their legal merits. Mr. Trump has also filed a defamation suit against the Wall Street Journal for reporting that he wrote a warm birthday note to Epstein on the occasion of the financier’s 50th birthday.

Mrs. Trump, meanwhile, has a formidable legal operation that for years has aggressively fought back at journalists exploring anything about her past that contradicts her official narrative, which will soon be captured in a documentary from Amazon. The company paid Mrs. Trump $40 million for her participation in the production.

Mrs. Trump sued the Daily Mail in 2017 over an article it published on her past. The case — brought in the U.K., where the standards for libel are more forgiving for plaintiffs — was settled for an amount that was reported to be about $3 million. The Daily Mail published a rare apology.

Mr. Biden’s invocation of Mr. Wolff spotlights an incongruity in Mrs. Trump’s legal strategy first observed by Oliver Darcy of “Status.” The first lady’s lawyers have in recent months brought legal pressure to bear on both the Daily Beast —  for an article another writer wrote about Mr. Wolff’s appearance on the podcast — and a Democratic strategist, James Carville, for airing Mr. Wolff’s claims.

Curiously, though, no pushback has been directed at Mr. Wolff himself, a frequent — and often withering — chronicler of Mr. Trump’s time in power. Mr. Darcy speculates that “even the act of serving him with a legal threat would become its own news cycle, amplifying the very allegations Melania’s team is trying to scrub from the public record.”

Mr. Wolff has been acting like he has nothing to fear from Mrs. Trump. On Friday, Mr. Wolff appeared on Instagram, sitting in his Hamptons house, and described an ongoing contretemps between Mrs. Trump’s legal team and the British publishers of a new book about Prince Andrew that, Mr. Wolff says, contains reporting about Mrs. Trump and Epstein. According to Mr. Wolff, the print edition of the book contains the section on Mrs. Trump, while the digital and audio versions of the book do not.

One possibility for why Mrs. Trump is refraining from moving against Mr. Wolff could center less on public relations and more on legal strategy. A suit against Mr. Wolff would provide the opportunity to request documents via discovery, and could enable the journalist to prove — in court — the veracity of his claims about how the first couple first came to know each other.