He claimed the panel was a "whitewash" and accused Ms Suu Kyi, his long-time friend, of lacking "moral leadership".
Myanmar said he should "review himself" over the "personal attack".
More than 650,000 Rohingya people, from a mostly-Muslim minority in Buddhist Myanmar, fled to Bangladesh last year in the face of a military crackdown.
Many are now living in refugee camps in the neighbouring country. Bangladesh has said they will all be returned to Myanmar within two years.
Mr Richardson added that Ms Suu Kyi had been "furious" when he raised the case of two Reuters reporters on trial in Myanmar.
The journalists have been charged with breaching the Official Secrets Act while working on coverage of the Rohingya crisis.
Ms Suu Kyi "exploded" at Mr Richardson when he mentioned the journalists, he told the New York Times.
"Her face was quivering, and if she had been a little closer to me, she might have hit me, she was so furious," Mr Richardson said.
Mr Richardson, a former adviser to the Clinton administration, has known Ms Suu Kyi for decades, and visited the Nobel laureate while she was under house arrest in the 1990s.
He told Reuters he was resigning from the advisory board because it was a "whitewash" and he did not want to be part of a "cheerleading squad for the government".
He was "alarmed by the lack of sincerity with which the critical issue of citizenship was discussed," he wrote in a statement.