Iran's judiciary denied the torture claims.
The killing of Hassan Turkman, a water company security guard, occurred during some of the worst unrest in a decade over economic hardships in Iran.
"In letters, Thomas Bach, the International Olympic Committee president, had made direct personal appeals to the Supreme Leader and to the President of Iran this week and asked for mercy for Navid Afkari, while respecting the sovereignty of the Islamic Republic of Iran".
Afkari had been found guilty of "voluntary homicide" for stabbing to death Hossein Torkman, a water department worker, on August 2, 2018, according to the judiciary.
Afkari's mother, in an emotional video, denounced the sham trial against her sons, which include Afkari's two other brothers who have also been arrested and face criminal charges.
The provincial court in Shiraz also sentenced Afkari's brothers Vahid Afkari and Habib Afkari to 54 and 27 years in prison, respectively, over the killing.
"I would just like to say that I, too, respectfully and humbly ask the government officials in Iran to please not execute this man and spare his life".
Facebook most common platform for anti-Palestinian racism in Israel
His comments came after Bahrain agreed to normalize ties with Israel as part of a diplomatic push by US President Donald Trump . It was captured by Israel in the 1967 war with Arab states but remains an area claimed by the Palestinians, who are Arabs.
Addressing the protests, Esmaeili said, according to the Tehran newspaper, that the campaign aimed at stopping the execution led the judiciary to provide a videotaped confession and re-enactments carried out by Afkari himself to Iranian State TV IRIB.
Wrestling is a hugely popular in Iran, one of the world's superpowers in the sport.
Paknia represents three men sentenced to death over links to similar protests in November, but the trio's execution was halted over a request to the supreme court to review the verdict.
"These brothers are the latest victims of Iran's deeply flawed criminal justice system, and their case is further evidence that Iranian courts systematically rely on "confessions" obtained under torture and other ill-treatment to secure criminal convictions, in contravention of global law", said Diana Eltahawy, Amnesty International's deputy regional director for the Middle East and North Africa.
Iranian rights activist Emaddein Baghi tweeted that Afkari's execution was a "great sin" as the judiciary should have tried to persuade the murdered man's family to forgive the wrestler.
UFC president Dana White wrote, "Iran, please spare Navid Afkari". "First, he's a human being".
The death sentence had been carried out at "the insistence of the victim's family", said Mousavi, the prosecutor general for Fars province.
"Hearing that Iran is looking to execute a great and popular wrestling star, 27-year-old Navid Afkarai, whose sole act was an anti-government demonstration on the streets", President Trump tweeted on September 3. "They were protesting the "country's worsening economic situation and inflation", Trump tweeted.