Regarded as the best batsman in the world, Smith admitted on Sunday to orchestrating a plan to ball tamper during Australia's third Test defeat by South Africa.
The 28-year-old has been banned for one Test and fined his entire match fee by cricket's world governing body
In a statement, the Royals said Smith had taken the decision to stand down so that the team can prepare for the IPL "without the ongoing distractions."
India's vice-captain Ajinkya Rahane will replace Smith.
Manoj Badale, Royals co-owner, said in a statement: "It is important that all cricket fans retain a balanced perspective on the situation.
"What happened in South Africa was clearly wrong, especially given that it appears to have been premeditated. That said, this will be a difficult time for Steve as well, given how much he cares about the game."
The IPL starts on Saturday, April 7, with the Royals set to play their first match of the season against Sunrisers Hyderabad on April 9.
Cameron Bancroft was caught on camera pulling tape from his pocket and rubbing it on the ball on day three of the third Test.
The method, used to cause the ball to swerve unpredictably in the air, goes against cricket's code of conduct and is considered illegal within the sport.
Smith has stepped down as Australia captain for the remainder of the Test match after admitting that the team's "leadership group" had a plan to tamper with the ball to "get an advantage."
The batsman said that the idea of tampering with the ball was first raised at lunch Saturday. He said head coach Darren Lehmann and the other coaching staff were not aware of the plan.
Bancroft was fined 75% of his match fee, warned for his part in the affair as well as being hit by three demerit points.
The 25-year-old, playing in just his eighth Test, told reporters he "panicked" when he realized his tampering had been spotted on camera, and shoved the yellow tape that he'd pressed against the ball down his trousers.
"I saw an opportunity to potentially use some tape to get some granules from the rough patches on the wicket and try to change the ball condition," Bancroft said. "It didn't work."
Rule breaking 'totally abhorred' by Australian public
John Buchanan, who coached the Australian cricket side from 1999 to 2007, says he expects Smith to receive further punishment.
"This one decision is a real clanger," Buchanan, who led his country to three successive World Cup wins, told CNN World Sport. "It's a massive decision that's been made.
"The consequences are that he will hopefully be given the opportunity to resign the captaincy, but if he doesn't take that option then it will be taken away from him and I would have thought for good."
Admitting that he would be in favor of Smith being stripped of the captaincy, Buchanan also added that the incident has damaged the relationship between Australia's revered cricket team and its public.
"The notion of Australians playing outside the rules to win games is totally abhorred," he says.
"What they've done here is lost the trust of the Australian public ... The captain of the team is second only to the prime minister as the cliché goes. Unfortunately, this decision has broken that relationship with the Australian community.
"It's not only about the cricket team, it's also about how Australians are viewed, how Australian sportspeople are viewed around the world."
Outrage from fans and former players
The scandal has caused outrage in Australia and prompted widespread calls for a full investigation from fans, former players, official sporting bodies and senior politicians, including Prime Minister Malcolm Turnbull.
The Australian Prime Minister said he woke up "shocked and bitterly disappointed" after hearing the news of what had unfolded. He said he had spoken to Cricket Australia chairman David Peever to discuss the incident.
Former Australian captain Michael Clarke, who expressed his anger on Twitter, later told Sports Sunday on the Nine network, that he believed Smith would be "sitting in his hotel room in tears."
"I can't believe the leadership group has made the decision to do this, that they've gone and got the young kid who's played is eighth test match to do that ... It's premeditated cheating, it's blatant cheating," he tweeted.
Peter FitzSimons, author and commentator, writing Sunday in an opinion piece in the Sydney Morning Herald, talked of the "murder of cricket.'
"The horror of what happened in South Africa overnight was just how our national cricketing leadership could engage in such cold-blooded, premeditated, clear-eyed CHEATING.
"It's wrong! It is not only against the spirit of the game, but is so far the other side of the laws of the game, it's nudging up against the murder of cricket."