Players hatched the idea of cheating with sticky tape, which they hoped would capture debris from the pitch and scuff one side of the ball, at lunch on Saturday.
"I won't be considering stepping down. I still think I'm the right person for the job," Smith told reporters.
Cameron Bancroft used the tape while working on the ball in the post-lunch session then attempted to hide it from umpires.
Bancroft has been charged with ball tampering by match referee Andy Pycroft but the heat is well and truly on the man behind the plan.
"Today was a big mistake on my behalf and on the leadership group's behalf as well," Smith said.
"I take responsibility as the captain. I need to take control of the ship.
"I'm incredibly sorry for trying to bring the game into disrepute the way we did.
"This is certainly something I'm not proud of and something that I can hope to learn from and come back strong from.
"My integrity, the team's integrity, the leadership group's integrity has come into question."
The skipper wouldn't divulge the other teammates involved in the discussion. The leadership group has featured David Warner, Mitchell Starc and Josh Hazlewood in the past.
Smith, with the exception of a couple of incidents on a spiteful tour of India last year, had enjoyed a squeaky-clean start to his century-laden captaincy.
"I am embarrassed," the 28-year-old said.
"I know the boys in the shed are embarrassed as well and I feel for Cam.
"It's a poor reflection on everyone in that dressing room, particularly the leaders.
"If we weren't caught I'd still feel incredibly bad about it ... it's a big error in judgment.
"It was a poor choice and deeply regrettable.
"I can promise you it won't happen again."
Smith insisted coaching staff, including Darren Lehmann, were not aware of the premeditated plan and that his side have never tried using tape to scuff the ball before.
"Obviously it didn't work. The umpires didn't see it change the way the ball was behaving," Smith said.
"We saw this game as such an important game.
"We've seen the ball reversing quite a lot throughout this series and our ball just didn't look like it was going to go."
Discussion about both sides attaining early reverse-swing has dominated the series.
Questions are already being asked about the tourists' conduct in previous Tests.
"You can ask questions as much as you like but I can promise you this is the first time it's happened," Smith said.
"It's not what we want to see in the game. It's not what the Australian cricket team's about."