Remembering his "very tearful" first encounter with Mandela, the nine-time major winner paused Friday to compose himself and hold back more tears.
Mandela, the beloved former South African president and Nobel laureate, died Thursday at 95.
Renowned as a fierce competitor on the golf course, Player was invited to meet Mandela at his office in Johannesburg after the anti-apartheid leader's release after 27 years in prison.
But he didn't expect to do what he did, Player said.
"I knelt down and I kissed his feet and I said, 'I have never kissed anybody's feet in my life,' and I said, 'I have so much admiration for you.' I said to him, 'It is remarkable, how can you not have revenge?'"
Mandela's reply, according to Player, was: "You have got to start a new life and forgive and go ahead."
Player, speaking at the Nedbank Golf Challenge at Sun City, said Mandela's ability to inspire with compassion left him amazed then — and still does.
"It was very tearful for me, because when you think of a man that has gone to jail for all those years for doing the right thing, not the wrong thing, it is hard to comprehend that a man can come out and be like that," Player said. "He was an exceptional man."
From that first meeting, Player and Mandela would cross paths regularly as the golfer, one of South Africa's greatest sportsmen, worked with the president on charity projects.
Once, Player remembered with a big smile, Mandela landed at a charity tournament in a helicopter to lend support.
"I had to meet him when the helicopter arrived and open the door. Now I had been around him all these years raising money for young black children and I opened the door, and he says 'Good morning Gary, do you remember me?'" Player recalled, imitating Mandela's unique rasping voice. "Just wonderful."
Although Player wasn't sure if he ever saw Mandela swing a golf club, he knew that the anti-apartheid leader "realized the value of sport" and even followed Player's career overseas while he was imprisoned by South Africa's former racist regime.
"He said to me, 'When I was in jail, I used to watch you playing.' He was very complimentary," Player said.