Unsworth is seeking more than $75,000 in damages, and demands that Musk - who recently called him a "child rapist" - stop making any further allegations.
Unsworth's comment prompted a series of tweets by Musk, including on July 15 when he said, "Sorry pedo guy, you really did ask for it". Musk and engineers from his company, SpaceX, had created a small submarine to help with the rescue.
Though Mr Unsworth told CNN three days later that Mr Musk's offer was a "PR stunt" that had no chance of working and that he could "stick his submarine where it hurts", he said that did not justify Mr Musk's use of Twitter and the media to defame him.
The tweet was deleted, but Mr Musk later insisted to journalists that Mr Unsworth had moved to Thailand "for a child bride who was about 12 years old at the time".
After initially apologising for his "pedo" accusation, the Paypal co-founder lashed out again in August, tweeting to his 22 million followers: "Don't you think it's unusual he hasn't sued me?"
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Some hours later, Unsworth's lawyer replied to the tweet with a picture of a letter sent to Musk informing him he would be sued for libel. "Mr Unsworth is not a child rapist".
An global team of divers and rescue workers - one of whom, a Thai Navy SEAL, died during the operation - ultimately brought the boys to the surface on July 10.
The 47-year-old billionaire, who earlier this month faced investor ire over smoking marijuana on a live web show, has indicated in the past that Tesla's customers may face a longer response time because of a significant increase in vehicle delivery volume in North America. Chiang Rai is renowned for child sex-trafficking. He started going to Thailand in 2011, where he explored and mapped caves, the documents said. He was the first foreign rescuer to arrive. Harper, who had just returned from exploring Thai caves with Mr Unsworth, brought two other divers, John Volanthen and Rick Stanton, to help with the rescue.
Unsworth started caving when he was 16 years old and was involved in several cave rescues in the United Kingdom, according to the lawsuit.
It conceded that Mr Unsworth was not involved in the final planning stages of the rescue because he did not have enough experience to make the dive.