"I am taking this action, after lengthy consideration, so both the institution and I can move forward," Barry Munitz said.
The J Paul Getty Trust also deals with art conservation and research.
This resignation is the latest problem for the trust, which, in 2005, returned artworks that were supposedly stolen from Italy.
The US Council on Foundations - the nation's leading group of charitable organisations - launched an investigation into the trust's financial affairs, after it was reported in the Los Angeles Times about alleged extravagant spending by Barry Munitz.
Last December the council placed the trust on a 60-day probation for failing to come up with acceptable documentation for its review.
Then last year the curator Marion True resigned a month before appearing in a court in Rome where she was charged with the traffic of stolen antiquities.