He said he had "spoken in anger" after Vern Unsworth ridiculed a mini-submarine he had commissioned for the rescue effort as a "PR stunt".
"His actions against me do not justify my actions against him, and for that I apologise," Mr Musk said.
Mr Unsworth told reporters on Monday he was considering legal action.
Twelve boys and their football coach were rescued from deep within a cave earlier this month after being trapped for more than two weeks.
Mr Unsworth's knowledge of the cave complex is said to have played a key role in the rescue effort.
He travelled into the caves in the first days after the boys went missing and helped bring in top international cave rescue experts for the mission.
Mr Musk, head of the SpaceX and Tesla companies, visited the command centre in Thailand during the rescue, announcing he had left the rescuers a mini-submarine.
What did Musk say?
In a reply to another user on Twitter on Wednesday, referring to an article about his involvement in the rescue mission, he wrote that he was apologising to both Mr Unsworth and the companies he led.
"The fault is mine and mine alone," he declared.
At the same time, he accused Mr Unsworth of telling "untruths" about his mini-submarine, which was not used in the rescue.
How did the row erupt?
Mr Unsworth had said the mini-sub would have had "absolutely no chance of working" - and suggested Mr Musk could "stick his submarine where it hurts".
The entrepreneur fired back with a now deleted tweet where, without using Mr Unsworth's name, he referred to a "British expat guy who lives in Thailand".
He said he would make a video showing the mini-sub making it deep inside the cave "no problemo", adding, "Sorry pedo guy, you really did ask for it."
The tweets, sent to his more than 22 million followers, were later deleted.
Asked if he would take legal action against Mr Musk, Mr Unsworth told AFP news agency: "If it's what I think it is yes."
No charges or complaints had ever been filed against Mr Unsworth, Reuters news agency quoted an unnamed police officer in the Chiang Rai area as saying.
Separately, Mr Musk attacked the Thai official in charge of the command centre, Narongsak Osotthanakorn, for saying the mini-sub had been "not practical", though "technologically sophisticated".
Mr Musk said Mr Osotthanakorn had been "described inaccurately as 'rescue chief'" and was "not the subject matter expert".
He also tweeted an earlier email from Richard Stanton - one of two British caving experts called in to front the rescue - who had urged the business leader to build the capsule as quickly as possible.
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