Indian cricket board president Sharad Pawar said that Chappell, 58, sent his office an e-mail saying he did not wish to renew his contract as coach for "family and personal reasons."
"I can confirm that Chappell does not want to stay as coach for family and personal reasons," Pawar, a federal minister, told reporters here.
The former Australian captain has served as coach since June 2005 and his two-year contract was due to end after the World Cup.
His resignation comes two days before the Board of Control for Cricket in India (BCCI) meets in Mumbai on Friday and Saturday to take stock of the World Cup debacle.
Sources told AFP that Chappell will attend the meeting to present his report on the team's dismal performance.
India were upset by Bangladesh and also lost to Sri Lanka to be knocked out in the preliminary stage of the ongoing World Cup in the Caribbean.
Chappell's e-mail to Pawar said he wished "Indian cricket much joy and success in the future," sources said.
"I am grateful to the players with whom I have worked for the challenges that they presented me with and which I tried to meet in a professional, methodical and interesting way in the interests of the team and the individual," Chappell wrote.
"I would like to make special mention of my support staff, without whom I would not have survived the rigours of the past 22 months.
"Ian Frazer (biomechanist) and Greg King (trainer) deserve special mention for their efforts, as do John Gloster (physiotherapist), S. Ramakrishnan (computer analyst) and Ramesh Mane (masseur).
"I am particularly grateful for the wonderful support of my family, especially my wife Judy, who has enjoyed the experience as much as I.
"I look forward to continuing my contact with India in the coming years and I wish Indian cricket much joy and success in the future," he wrote.
The outspoken Chappell generated more heat, controversy and headlines in two years as coach than his genial predecessor John Wright did in five years in the job.
Chappell became the media's whipping boy early in his tenure when criticism of Sourav Ganguly's laid-back attitude saw the country's most successful captain being thrown out of the team.
Chappell's displeasure at the exclusion of juniors such as Suresh Raina for the World Cup, as revealed in a text message to a journalist, raised the heckles of both the selectors and senior players.
"If Greg had his way, even Sachin Tendulkar would have been dropped," the Kolkata-based Telegraph newspaper quoted an unnamed selector as saying.
In hindsight, Chappell may have been right since neither Tendulkar, one-day cricket's most successful batsman, nor the other experienced players managed to take India past the first round.
Influential cricket officials believe the team has not moved forward in the two years that Chappell has been coach. Many say the 2003 World Cup finalists have fallen behind other teams.
Chappell's resignation came on the day senior pro Tendulkar responded to media reports that Chappell had blamed the unhelpful attitude of senior players for the disastrous World Cup campaign.
In a rare public outburst, the normally reticent Tendulkar, 33, said he would be hurt if Chappell had questioned his or the team's commitment.
"I've given my heart and my soul for 17 years," said Tendulkar, statistically the most successful batsman ever with a world record 35 Test and 41 one-day centuries
"No coach had mentioned even in passing that my attitude was not correct," he told the Times of India, which ran his comments as the lead front-page story.
"It's not that we are defending ourselves. We do realise that we played badly and, as a team, we take full responsibility for that. But what hurt us most is if the coach has questioned our attitude."
India need to find Chappell's replacement quickly since they are due to tour Bangladesh in May for two Tests and three one-day internationals.