Lochte, who flew to the United States the day after Sunday's incident, said he should have been more careful and candid in his account, but said it had been traumatic to have a stranger point a gun at him in a foreign country and demand money.
The 32-year-old, one of America's most decorated swimmers and the most outspoken about the incident, had originally said he and three team mates, including Gunnar Bentz and Jack Conger, were stopped in a taxi on the way back from a party by gunmen posing as police who stole $400 from them.
However, police called this a lie, and Rio's mayor said on Friday he had "pity and contempt" for the swimmers.
They said one of the swimmers had vandalized a gas station after the group stopped the taxi there to use a bathroom. The swimmers started an argument with staff at the station, who demanded payment for the damage, police added.
After security video emerged of the incident, the U.S. Olympic Committee admitted an act of vandalism had taken place and apologized for the incident, which had embarrassed the host city, angered the police and government, and dominated news coverage of South America's first Olympics.
"I want to apologize for my behavior of last weekend," Lochte said on his Instagram account.
"Regardless of the behavior of anyone else that night, I should have been much more responsible in how I handled myself and for that I am sorry to my teammates, my fans, my fellow competitors, my sponsors and the hosts of this great event."
'PITY AND CONTEMPT'
Police said Feigen, the last of the four swimmers still in Brazil, had agreed at a hearing with a judge on Friday to pay 35,000 reais ($11,000) to a sporting charity after giving false testimony.
The charity, the Reaction Institute, said Feigen's lawyer had already contacted them about making the payment. The institute brings sports to low-income communities and helped train Brazilian gold medalist judoka Rafaela Silva.
A police source said Feigen's passport would be released once he made the payment and presented a receipt to authorities. Police still want to speak to Lochte, who was also summonsed to appear before the judge on Friday, and say they hope to agree with U.S. authorities to interview him in the United States.
Bentz and Conger arrived in Miami on Friday aboard an American Airlines flight from Brazil, having sat in curtained-off seats for much of the journey. Police had pulled them off a flight on Wednesday night for further questioning.
Bentz and Conger said they were not robbed in revised testimony given to police hours before their departure, senior Rio police officer Alexandre Braga told Reuters. Instead, they said Lochte had vandalized a local gas station early on Sunday and had an altercation with a security guard.
The three other swimmers in the car tried to leave as quickly as possible because they feared Lochte would cause more damage, G1 said, quoting from the revised testimony.
Bentz and Conger were jeered by angry Brazilians before leaving for Rio airport on Thursday, but received support from some other passengers on their flight .
"I've got boys that age," said Lee McNutt, a 61-year-old Silicon Valley salesman returning from the Games.
"I just said 'don't let this get you down. I don't think real sports fans are going to worry about this. You're young and this will pass. I told them to take tonight off. Tomorrow, start training for Tokyo."
In Miami, the two swimmers caught a connecting flight, walking through the airport in matching backpacks, a Reuters witness said. Conger kept headphones on and wore an orange Team USA hooded sweatshirt.
Many Brazilians have reacted angrily to the news that the swimmers fabricated their story, but others have said it shows the underlying problems in their crime-ridden nation.
"If this was a Brazilian or it wasn't the Olympics, no one would have worried about it," said Janete Carvalho, 54, an English teacher, said on the streets of downtown Rio.
"It wouldn't have cost them anything to have told the truth about what happened... I don't think it will harm Brazil's image, only theirs."
The incident followed a series of muggings and armed robberies of high-profile athletes and visitors in Rio, including two government ministers.
On Thursday, the USOC apologized "to our hosts in Rio and the people of Brazil" and said the swimmers' behavior was unacceptable.
It said it would further review the incident and assess any potential consequences for the athletes. USA Swimming said it could also take possible action.
Rio Mayor Eduardo Paes on Friday accepted the apology given by the U.S. Olympic Committee and said the swimmers' behavior did not represent the good spirit of Team USA.
But he added: "My only feelings toward them (the swimmers) are pity and contempt."