President Sebastian Pinera of Chile has met the 33 rescued miners at the hospital where they are recovering and announced a review of mine safety.
Speaking outside Copiapo Hospital, he promised that never again in Chile would people be allowed to work in such inhumane conditions.
The president described the rescue of the men after more than two months trapped underground as "a miracle".
At least two of the men are due to be discharged from hospital on Thursday.
Of the rest, some are receiving treatment for dental and eye problem and two have the lung disease silicosis.
The rescue concluded earlier when Luis Urzua was the last trapped man to be brought to the surface.
Mr Urzua emerged to jubilant scenes at 2155 local time on Wednesday (0055 GMT on Thursday), ending a rescue operation of more than 22 hours.
As he left the capsule he was congratulated by President Pinera, who met him again during a visit to Copiapo hospital on Thursday and told reporters of a conversation they had shortly after the mine supervisor had finally been rescued.
The Chilean leader said that Luis Urzua's words had "touched my soul... when he said that this should never happen again".
Mr Pinera said it was impossible to guarantee that Chile would never face such an accident in the future.
"But we can guarantee one thing: never again in our country will we allow working in conditions so inhumane and so unsafe as happened in the San Jose mine and many other places in our country."
President Pinera promised a review that would lead to a "very radical change" affecting the health and safety of workers in the transport, fishing and construction industry as much as mining.
All of the miners are now in hospital in Copiapo, being treated in two wards that have been darkened to allow the men to adjust to the light.
The BBC's Rajesh Mirchandani, outside the hospital, says the eldest miner, Mario Gomez, 63, is being treated for pneumonia and the lung disease silicosis. The second man out of the mine, Mario Sepulveda, also has silicosis.
At a news conference on Thursday, the hospital's medical sub-director, Dr Jorge Montes, told reporters that the condition of all 33 men was under control.
Some of them had received dental surgery the previous day and further dental treatment would be provided to others, he added.
Seventeen of the miners are being treated in a special unit of the hospital while the other 16 are in a general section. Each miner is being allowed up to three visits per day.
The miners had been told they would need to be held in the hospital for 48 hours but because of the surprisingly good health of most of the men, at least two of them will be discharged in a matter of hours.
"We estimate that about 1600 hours [1900 GMT] this afternoon some of them are going to leave this hospital," Dr Montes said, adding that at least two or three would be discharged.
"All the patients have completed their examinations. But the people who are going to be released are going to be the ones with the lowest index of complications. The three patients have evolved very positively."
Health Minister Jaime Manalich said it had come as a shock that the miners were in such a good condition, and that many of them were in a position to be discharged over the next 24 hours.
"Even when we recognise the efforts on the part of the medical team we were completely surprised and we call this a miracle, because any effort we could have made doesn't explain the health condition that these people have today," he added.
One of the children, Ximena, has described how her father has become so pale after his 69 days underground that he resembles the cartoon ghost, Casper.
"In general, he's in good spirits. But then when he remembers the first few days after the accident, he starts crying, he gets very upset," she wrote.
"But then he pulls himself together and his spirits are high once more."
During the president's visit to the hospital, he posed with the miners for a group photograph and invited them for a game of football later this month against the officials who had helped rescue them.
Mr Pinera later cracked a joke with reporters, suggesting that the winners would be allowed to live at the presidential palace, La Moneda, while the losers would have to go back down the mine.
There were scenes of celebration at the mine, in Copiapo and around Chile when the long rescue drew to a close on Wednesday night.
Thirty-three balloons in Chilean colours were released above the mine after the capsule carrying Mr Urzua up the 624m (2,047ft) rescue shaft emerged to songs and applause.
Hours later, relatives of the rescued miners began to pack up their belongings at Camp Hope, the impromptu settlement at the San Jose mine that had become home for families and journalists.
Chile's Mining Minister Laurence Golborne returned to the mine on Thursday to tell reporters that two of the three rescue shafts, Plan A and C, would be sealed with concrete while the Plan B shaft which eventually reached the miners would be sealed provisionally.
Like President Pinera, Mr Golborne drew attention to the safety of Chile's industrial workers, pointing out that no accidents had taken place during the 70-day operation to free the miners.
He said everyone involved was aware they could not afford an accident and had taken responsibility for safety.
Avoiding such incidents, he suggested, was not just a matter for legislation: "The real origin to avoid this in the future lies in the consciousness of people: of workers, entrepreneurs... all society to face situations of safety at work."
The miners were trapped when part of the San Jose copper and gold mine in the Atacama desert collapsed on 5 August. It was 17 days later when rescuers discovered the men were alive.
The last man to leave the mine was rescuer Manuel Gonzalez, who returned to the surface just under two-and-a-half hours after Mr Urzua.
He also returned to the site on Thursday, describing what had happened as an "unforgettable experience".
Messages of congratulations to the Chilean people have poured in from around the world - from US President Barack Obama, Pope Benedict XVI, UK Prime Minister David Cameron and many others.
UN Secretary General Ban Ki-moon added his congratulations in a telephone call to President Pinera on Thursday, a UN spokesman said.
"The secretary general joined with the people of Chile and the families of the heroic miners to celebrate what he called the extraordinary triumph of human ingenuity and the strength of the human spirit," Martin Nesirky said.