Mr Musk's comments highlighted concerns about the platform's future after he pledged it would not become a "free-for-all hellscape."
The self-declared "free speech absolutist," was responding to former Democratic presidential candidate Hillary Clinton, who tweeted about Republicans who promote baseless conspiracy theories and the violent attack on Paul Pelosi in San Francisco on Friday.
"There is a tiny possibility there might be more to this story than meets the eye," Mr Musk told Ms Clinton, attaching a link to the story, which is no longer accessible, by the conservative Santa Monica Observer.
The weekly has published conspiracy theories in the past, including that a body double for Ms Clinton was sent to a debate with Donald Trump during the 2016 election campaign, according to the Los Angeles Times.
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Mr Musk later deleted his tweet after it swiftly became a focal point for critics who have been nervous about the direction in which he intends to take Twitter.
His outspoken and controversial tweets have courted trouble in the past, and he has pledged to reduce content moderation, relying more on computer algorithms than human monitors.
Conservatives say past moderation has unfairly attacked their views.
In a message meant to reassure Twitter advertisers about his leadership, Mr Musk said this week that he realised Twitter "cannot become a free-for-all hellscape where anything can be said with no consequences."
But detractors warn that without standards, the world's "digital town square" is at risk of becoming flooded with misinformation, with possibly perilous consequences for democracy and public health.
"Clinton: Conspiracy theories are getting people killed and we shouldn't amplify them. Owner of Twitter: But have you considered this conspiracy theory?" wrote University of Denver political scientist Seth Masket after Musk's Sunday tweet.
Ms Pelosi, who is second in line to the US presidency, has said her family is "heartbroken and traumatised" after the intruder broke into the couple's San Francisco home early on Friday and attacked her husband with a hammer, fracturing his skull.
Mr Pelosi, 82, is recovering in hospital.
President Joe Biden has said it appeared the assault was "intended for Nancy," and called out increasingly polarising political speech.
"The Republican Party and its mouthpieces now regularly spread hate and deranged conspiracy theories. It is shocking, but not surprising, that violence is the result," Ms Clinton said in her tweet.
Mr Musk's response came hours after Twitter said the site was being hit by a trolling campaign testing its moderation policies under the billionaire's leadership.
"Twitter's policies haven't changed ... And we're taking steps to put a stop to an organised effort to make people think we have," tweeted the platform's chief of safety and integrity, Yoel Roth.