Smith was stripped of the Australian captaincy on Sunday and suspended for one test match by the International Cricket Council (ICC) after admitting conspiring to tamper with the ball during the third test against South Africa in Cape Town.
He was due to captain Royals in this year's IPL but has been replaced by India's Ajinkya Rahane.
In a statement, the Royals said Smith believes "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."
Smith re-joined Royals for this year's competition having previously played for them between 2014-15, with the side having paid $1.07 million for his services.
"The incident in Cape Town has certainly disturbed the cricketing world," Zubin Bharucha, the franchise's head of cricket, said.
"We have been in constant touch with the BCCI (Board of Control for Cricket in India) and taken their counsel. Furthermore we have been in regular contact with Steve."
Smith confessed on Saturday that he and a leadership group in the Australia team had hatched a plan at lunch on the third day of the test at Newlands to alter the condition of the ball.
Australia's opening batsman Cameron Bancroft, who was caught on camera attempting to carry out the plan, was handed three demerit points and a 75 percent fine of his match fee but escaped a ban.
"We agree with Steve's decision to step down as captain, and we are fortunate to have such a capable successor with Ajinkya Rahane," said Manoj Badale, the co-owner of Rajasthan Royals.
"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."
(Reporting by Sudipto Ganguly; editing by Christian Radnedge)
