Video suggests Wagner leader visited Belarus to address his troops this week.

President Cyril Ramaphosa announced that Mr. Putin had, by “mutual agreement,” decided not to attend the summit in person, and would send his foreign minister, Sergey V. Lavrov, in his place. Russian state media said that Mr. Putin would participate via videoconference in the summit, a long-planned meeting of the heads of state of Brazil, Russia, India, China and South Africa, a bloc known as BRICS.

While this decision eases South Africa’s immediate dilemma, the country is still walking a shaky and very public tightrope as it tries to maintain strong ties with each of its superpower allies when they are at odds with one another.

South Africa has faced withering criticism from the United States for refusing to condemn Russia’s invasion of Ukraine. American officials have in addition accused South Africa of providing arms to Russia, a claim that the government has denied and that Mr. Ramaphosa said was being investigated.

Critics at home have accused Mr. Ramaphosa, who faces a tough re-election contest next year, of taking a soft stance toward Russia that could hurt South Africa economically. American lawmakers and government officials have suggested that the U.S. should consider revoking trade benefits for South Africa and rethink the alliance between the countries all together. Hosting Mr. Putin would only have inflamed those demands.

Mr. Putin is the subject of an arrest warrant by the International Criminal Court, which accuses him of being responsible for the abduction of Ukrainian children and their deportation to Russia. As a signatory to the court, South Africa would have been required to arrest the Russian president if he set foot on its soil.

Yet Mr. Putin had for months insisted that he would attend the summit in person, rejecting entreaties to stay home or attend by video. But he softened his stance after the instability set off last month by the brief revolt organized by the leader of the Wagner network, Yevgeny Prigozhin, according to a South African government official who requested anonymity to discuss internal deliberations.

Mr. Putin “became easier to persuade as a result of the recent domestic problems he is having,” said the official.

A spokesman for Mr. Ramaphosa, Vincent Magwenya, said he was unaware of whether the revolt had influenced Mr. Putin’s decision but that it was the result of lengthy deliberations.

South African officials have said in recent months that they feared that the question over Mr. Putin’s attendance at the BRICS meeting threatened to overshadow the agenda. BRICS has fashioned itself as an alternative to a world order centered on the U.S. and Europe, and a voice for nations that are not among the world’s superpowers.

BRICS has pushed for more developing countries to have seats on the U.N. Security Council, for rich nations to provide more funding to developing countries to address climate change, and for more equitable distribution of Covid-19 vaccines.

As the bloc’s newest and smallest member, South Africa is trying to wield more influence globally and fashion itself as the voice of Africa, analysts say.

South African officials have accused Western nations of having a double standard for calling to arrest Mr. Putin for war crimes in Ukraine, while escaping action by the international criminal court over the invasion of Iraq and Afghanistan.

Mr. Ramaphosa’s political party, the African National Congress, said as recently as Wednesday morning that it wished that Mr. Putin would attend the summit. But the party applauded the ultimate outcome. It will “let the BRICS summit focus on the pressing issues in the geopolitical situation,” said Mahlengi Bhengu, the A.N.C.’s national spokeswoman, in a news briefing on Wednesday.

While many who wanted Mr. Putin to attend may be disappointed, she said, “I do think that wisdom may have prevailed amongst our leaders.”

Mr. Ramaphosa had warned in a court affidavit made public on Tuesday that his country could suffer severe consequences if it arrested Mr. Putin. Russia “has made it clear” that an arrest “would be a declaration of war,” Mr. Ramaphosa said in the 32-page affidavit.

The Kremlin denied having made any direct threats toward South Africa, but on Wednesday, its spokesman, Dmitri S. Peskov, told reporters that “it’s absolutely clear to everyone what an attempt to encroach on the Russian leader means.”

South Africa’s largest opposition political party, the Democratic Alliance, had asked a court in Pretoria, the nation’s executive capital, to force the government to arrest Mr. Putin if he attended the summit, scheduled for Aug. 22 to 24.

The leader of the alliance, John Steenhuisen, praised Wednesday’s announcement.

“It averts a potential international crisis,” he said.

In 2015, South Africa faced international condemnation when it refused to arrest the then-president of Sudan, Omar Hassan al-Bashir, who was wanted by the international court on charges of war crimes and genocide arising from atrocities in the western province of Darfur. South Africa permitted Mr. al-Bashir to fly in and out of Johannesburg unimpeded for a meeting of the African Union. He is still wanted by the court.

Lynsey Chutel contributed reporting from Johannesburg, and Ivan Nechepurenko from Tbilisi, Georgia.