Wagner Group leader Yevgeny Prigozhin said on Monday that he did not want to overthrow President Vladimir Putin's government, after his aborted mutiny in Russia.
Mr Prigozhin intended to register a protest against Russia's ineffectual conduct of the war in Ukraine, he said.
He explained the reasons for launching the Wagner Group's march on Moscow in an 11-minute audio message posted on the Telegram messaging app.
“We went as a demonstration of protest, not to overthrow the government of the country,” Mr Prigozhin said in the message.
“Our march showed many things we discussed earlier: the serious problems with security in the country.
“In Russian towns, civilians met us with Russian flags and the symbols of Wagner … they were all happy when we passed through.”
Mr Putin on Monday said that he gave an order to avoid bloodshed during the weekend rebellion, saying the West and Kyiv wanted Russians to "kill each other".
"From the start of the events, on my orders steps were taken to avoid large-scale bloodshed," he said in a televised address, thanking Russians for their "endurance and unity, and patriotism".
"It was precisely this fratricide that Russia's enemies wanted: both the neo-Nazis in Kyiv and their western patrons, and all sorts of national traitors. They wanted Russian soldiers to kill each other.
"Civilian solidarity showed that any blackmail, any attempts to organise internal turmoil, is doomed to fail," Mr Putin said.
He said fighters employed by the mercenary outfit Wagner could join the military or leave for Belarus.
"Today you have the possibility to continue serving Russia by entering into a contract with the Ministry of Defence or other law enforcement agencies, or to return to your family and close ones," Mr Putin said.
"Whoever wants to can go to Belarus."
On Saturday, Wagner mercenaries, under Mr Prigozhin's direction, seized control of a military base in southern Russia and began to advance on Moscow.
A deal with the Kremlin led to the mission being aborted before troops reached the capital.
Mr Prigozhin repeated his frequent claim that Wagner was the most effective fighting force in Russia “and even the world”.
He said they should have been used from the start of Russia's invasion of Ukraine on February 24 last year.
Mr Prigozhin praised his troops for the way they had seized the southern Russian city of Rostov-on-Don without bloodshed and sent an armed convoy to within 200km of Moscow.
“We showed a master class, as it should have been on February 24, 2022," he said. "We did not have the goal of overthrowing the existing regime and the legally elected government.
“The aim of the march was to avoid the destruction of Wagner.”
Wagner chief and his fighters cheered by locals as they leave Rostov-on-Don
Western leaders said Russia emerged weakened after the day-long march across the country by rebels from the front line in Ukraine.
US President Joe Biden said he was briefed hour by hour by his national security team and had spoken with allies on a video conference.
“We made clear that we were not involved. We had nothing to do with it,” Mr Biden said. “This was part of a struggle within the Russian system.
“We're going to keep assessing the fallout of this weekend's events and the implications for Russia and Ukraine. But it's still too early to reach a definitive conclusion about where this is going.”
He also said he was in contact with Ukrainian President Volodymyr Zelenskyy.
“No matter what comes next, we'll keep making sure that our allies and our partners are closely aligned in how we are … responding to the situation,” Mr Biden added.
British Foreign Secretary James Cleverly has called Mr Prigozhin’s mutiny an “unprecedented challenge to Putin's authority”.
Mr Cleverly told the UK Parliament that it was “clear that cracks are emerging in Russians’ support for the war”.
Mr Prigozhin had driven a “coach and horses through President Putin's case for war” with his comments stating that the invasion was illegitimate.
“One of Putin's prodigies has publicly destroyed his case for the war in Ukraine,” Mr Cleverly said.
“The events of this weekend an unprecedented challenge to Putin's authority with an armoured column approaching his own capital city.”
Former prime minister Liz Truss urged the British government to make a plan “in the case of the implosion of Russia” and to ensure that Ukraine’s Nato membership application was fast-tracked.
Mr Cleverly also told MPs that any action against Ukraine by Belarus, where Mr Prigozhin is thought to have fled, “would be met with severe repercussions from the United Kingdom”.
There was no word about the revolt from Mr Putin, who had said on Saturday that the rebellion put Russia's very existence under threat while vowing to punish those behind it.
The Kremlin released a video of the President congratulating participants in an industrial forum, although there was no indication of when it had been filmed.
Mikhail Mishustin, who leads Mr Putin's cabinet as his Prime Minister, acknowledged that Russia had faced “a challenge to its stability” and called for public loyalty.
“We need to act together, as one team, and maintain the unity of all forces, rallying around the President,” Mr Mishustin told a televised government meeting.
