Musk had begun the exchange by responding to a tweet from WhatsApp co-founder Brian Acton on the deletefacebook hashtag.
The hashtag gained prominence after the world's largest social network upset users by mishandling data, which ended up in the hands of Cambridge Analytica, a political consultancy that worked on US President Donald Trump's 2016 election campaign.
"What's Facebook?" Musk tweeted on Friday.
Many users also urged the billionaire to delete the profiles of his companies on Facebook's photo-sharing app Instagram.
"Instagram's probably ok ... so long as it stays fairly independent," Musk responded.
"I don't use FB & never have, so don't think I'm some kind of martyr or my companies are taking a huge blow. Also, we don't advertise or pay for endorsements, so ... don't care."
Musk has had run-ins with Facebook founder Mark Zuckerberg in the past.
Last year, a war of words broke out between Musk and Zuckerberg over whether robots will become smart enough to kill their human creators.
When Zuckerberg was asked about Musk's views on the dangers of robots, he chided "naysayers" whose "doomsday scenarios" were "irresponsible".
In response, Musk tweeted: "His understanding of the subject is limited."
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