Shares of the electric carmaker skidded more than 10% on Friday amid concerns that the alleged sexual misconduct and Musk's political comments could threaten to damage Tesla's brand and sales.
Business Insider reported on Thursday that Musk's private company SpaceX paid $250,000 in 2018 to settle a sexual harassment claim from an unnamed private jet flight attendant who accused Musk of exposing himself to her.
The article quoted an anonymous person who said she was a friend of the flight attendant. The friend had provided a statement as part of the private settlement process, according to the article.
"I have a challenge to this liar who claims their friend saw me 'exposed' - describe just one thing, anything at all (scars, tattoos, ...) that isn't known by the public. She won't be able to do so, because it never happened," Musk wrote on Twitter late on Thursday.
Reuters was not able to verify the Business Insider account.
Musk and SpaceX did not respond to Reuters requests for comment on the Business Insider story or on tweets by the billionaire.
In a statement to Reuters on Friday, Business Insider said: "We stand by our story which is based on documents and interviews and which speaks for itself."
The sexual misconduct claims "have the potential from a PR perspective to turn people off," said Guidehouse Insights analyst Sam Abuelsamid.
Friday's share drop knocked about $66 billion off Tesla's market value and put the stock at its lowest since last August.
The shares have fallen sharply since early April as Musk's plan to buy Twitter sparked concerns about distraction and as China's COVID lockdown measures hurt production at Tesla.
Ross Gerber, CEO of Tesla investor Gerber Kawasaki Wealth Management, said he is concerned that Tesla relies so heavily on a single person, Musk, but he believes his liberal clients will continue buying Musk's cars, even if they don't like what he says, since Tesla's mission is clear.
Business Insider quoted the friend of the flight attendant as saying that in addition to allegedly exposing himself, Musk rubbed the flight attendant's thigh and offered to buy her a horse if she would "do more" during an in-flight massage.
The flight attendant came to believe that her refusal to accept Musk's proposal had hurt her opportunities to work at SpaceX and prompted her to hire a lawyer in 2018, according to Business Insider.
In 2018, the rocket company paid $250,000 in exchange for a promise not to sue over the claims and included a nondisclosure agreement that prevented the flight attendant from speaking about it, Business Insider said. The news site did not name the friend or the flight attendant.
Musk, who is in the midst of a contentious effort to buy Twitter Inc, said on Wednesday that he would vote Republican instead of Democrat, predicting a "dirty tricks campaign against me" would follow.
On Thursday evening, Musk first tweeted: "The attacks against me should be viewed through a political lens - this is their standard (despicable) playbook - but nothing will deter me from fighting for a good future and your right to free speech."
In the initial tweet, he did not specifically mention the allegations in the Business Insider article.
He also tweeted that the article was meant to interfere with the Twitter acquisition.
(Reporting by Hyunjoo Jin and Joey Roulette, Additional reporting by Ross Keber, Noel Randewich; diting by Frances Kerry, Bill Berkrot and Daniel Wallis)