Mr Unsworth said he was considering legal action, while some investors in Mr Musk's company had demanded an apology.
Mr Musk's spat with Mr Unsworth started last week, after rescue teams rejected the tech billionaire's offer of a mini-submarine created by his rocket company SpaceX to help extract the boys and their coach from the labyrinth of partly flooded passages.
The diver told reporters that he was seriously considering taking the legal route to fight his cause.
Another report from The Guardian stated that Musk has been facing a lot of scrutiny lately for his rants on Twitter and his aggressive attitude towards journalist and other critics.More news: 'Defended a Tyrant': McCain, GOP Lawmakers Rip Trump Over 'Tragic' Putin Summit
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During the interview, the diver railed against Musk and his submarine.
Writing on Twitter on Wednesday, Musk said his words against Unsworth had been "spoken in anger after Mr. Unsworth said several untruths & suggested I engage in a sexual act with the mini-sub".
Mr Musk said Mr Osotthanakorn had been "described inaccurately as "rescue chief" and was "not the subject matter expert".
He told financial analysts at a Tesla earnings call this year that their "boring bonehead questions" were "not cool". In April, Tesla's stock fell 5.16% after Musk mocked analysts who were anxious about the company's financial health.
Backlash to Musk's tirade against Unsworth has been fierce.
Musk's outburst drew criticism from investors in his company Tesla.
Musk's Sunday tweet, later deleted, had sent investors away from Tesla stock, which fell almost 3 percent Monday but recovered 4.1 percent on Tuesday. Musk's creation went unused, with authorities explaining that it was not a good fit for the mission.