James said her relationship with Ukip had become "increasingly difficult" and that she had been offered no support by the executive during her leadership.
She stood down as leader last month after just 18 days in charge.
James said she would continue in the European Parliament as an independent and would continue to serve her south east England constituency "effectively and diligently".
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But Farage, the party’s interim leader, told this website, "This is yet another act of irrational selfishness from Diane James. This pattern of behaviour says that is she unfit to continue as an MEP.”
He added, “She should do the honourable thing and resign."
In a statement, James said Parliament President Martin Shultz had accepted her request to stand as an independent.
The statement read, “At a high profile public event in Cambridge last week, I was asked why I had not completed the process to become leader of Ukip.
“I had little option, but to give the truthful response that, although nominated leader by popular vote in the membership, I found that I had no support within the executive and therefore no ability to carry forward the policies on which I had campaigned.
"My decision to retire from the election process and not complete it was very difficult personally and professionally, given that Ukip has dominated my life and all my efforts for over five years.
"In recent weeks, my relationship with the party has been increasingly difficult and I feel it is now time to move on."
The 24-strong contingent of Ukip MEPs elected in 2014, the biggest contingent from the UK, has been reduced to 20, with Amjad Bashir and Janice Atkinson also having left the party.
Ukip's new leader will be announced on 28 November.
Paul Nuttall, the frontrunner, Suzanne Evans and John Rees-Evans are competing for the job.
Meanwhile, the parliamentary group dominated by Ukip, the EFDD, has been told to repay €172,655 after Parliament ruled it had breached party financing rules.
It comes after an audit into the Alliance for Direct Democracy in Europe's spending was leaked last week.
ADDE, which was founded by Ukip in 2014, attacked Parliament's verdict and said it would take the matter to the European Court of Justice.