Smith said he had taken the decision “in the best interest” of the IPL franchise ahead of the new season, which starts next week.
The 28-year-old’s future as leader of the national team is under intense scrutiny as investigations continue into events during Australia’s third Test defeat against South Africa in Cape Town.
Smith has already been banned from the fourth Test, which begins on Friday, by the International Cricket Council after admitting trying to change the condition of the ball.
The scandal came to light on Saturday after Cameron Bancroft was caught on camera hiding sticky tape, which he had used to manipulate the ball with debris from the pitch, down his trousers.
Smith will be replaced as Royals captain by Ajinkya Rahane.
Royals head of cricket Zubin Bharucha said: “The incident in Cape Town has certainly disturbed the cricket world.
Steve believes given the current circumstances, ‘It’s in the best interest of the Rajasthan Royals that he steps down as captain so the team can get ready for the start of the IPL without the ongoing distractions’.
Co-owner Manoj Badale said: “Rajasthan Royals will do everything possible to protect the values and the integrity of the game.
“We agree with Steve’s decision to step down as captain, and we are fortunate to have such a capable successor with Ajinkya Rahane.
“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 pre-meditated.
“That said, this will be a difficult time for Steve as well, given how much he cares about the game.”
Cricket Australia is conducting an urgent investigation. Chief executive James Sutherland is due to arrive in South Africa today, while head of integrity Iain Roy and Pat Howard, head of team performance, are also due to land in Cape Town.
Sutherland said: “We know Australians want answers and we will keep you updated on our findings and next steps, as a matter of urgency.”
Cricket Australia said all of the Test squad are in South Africa to help with the investigation, despite reports Smith had flown home.
A statement read: “Cricket Australia can confirm that all members of the touring Test squad currently remain in South Africa.
At this stage, players will remain in South Africa to assist CA with inquiries.
Meanwhile, the MCC, guardians of the laws of the game and the spirit of cricket, have called for governing bodies to be more assertive with regards to on-field conduct.
MCC assistant secretary John Stephenson said: “The behaviour of some of the players in the current South Africa/Australia series, and other incidents in recent times in the game we all cherish, has fallen well below the standard required to inspire future generations of cricket-loving families.
“The time has come for a major shift in attitude and culture of all those with responsibility for leadership within the game, to give young players the kind of role models who will uphold standards, preserve cricket, and, vitally, the spirit of cricket for future generations.”