Smith and batsmen David Warner and Cameron Bancroft have been banned by Cricket Australia for the incident in the third Test against South Africa.
“It was a failure of my leadership,” said the 28-year-old at a news conference in Sydney.
Bancroft and Warner have also apologised for their actions.
Smith added: “I am deeply sorry, I love the game of cricket, I love kids wanting to play the great game of cricket that I love.
“Any time you think about making a questionable decision, think about affecting your parents.
“To see the way my old man has been, and my mum, it hurts. I just want to say I’m sorry for the pain I have brought to Australia and the fans and the public.”
In a social media post, Warner said the incident had “damaged cricket” and was “a stain on the game we all love”.
Bancroft spoke to the media in Perth and said: “It’s something I will regret for the rest of my life. I feel like I’ve let everyone down in Australia.”
The 25-year-old, who was playing in just his eighth Test, received a nine-month ban from international cricket after it was revealed he used sandpaper to damage the ball during the third Test against South Africa. He had initially claimed he used tape after the incident was caught on camera.
“I lied. I lied about the sandpaper,” Bancroft said. “I panicked in that situation and I’m very sorry. I feel like I’ve let everyone down in Australia.”
Smith admitted that the Aussie “leadership group” had devised a plan to tamper with the ball, and Cricket Australia’s investigation found Warner instructed Bancroft how to carry it out.
The pair have both been given year-long suspensions from international cricket.
Smith, 28, and Bancroft have also been banned from captaining Australia for at least the next two years, while Warner, 31, will not be considered for any team leadership positions in the future.
All three players have been sent home fro