Fetzer told Steele that he has "prayed with him and hurt with him over the past few weeks."
A lightning rod for criticism since he was elected chairman in 2009, Steele came under renewed scrutiny amid evidence of lavish spending by the committee, including a nearly $2,000 bill at a sex-themed nightclub in Los Angeles.
"More than ever, America needs the Republican Party to be a force for reform, transparency, and ethics in government. If we are going to be an effective agent for reform in America, we must first reform our party," said Fetzer in his letter to Steele.
Fetzer said he did not come to the decision to ask for Steele's resignation lightly, but that - in his personal opinion - the position has become untenable.
The spending controversy, particularly the idea of GOP dollars going to a Hollywood club featuring simulated bondage and lesbian sex, led RNC executive director Ken McKay to step down and Alex Castellanos, a prominent GOP pollster, to also suggest that Steele resign as chairman.
Sean Mahoney, a businessman and RNC member from New Hampshire, resigned from the RNC, blaming "the out-of-touch, free-spending culture of Washington" for infecting the Republican Party.
But Ron Nehring, an RNC member from California, dismissed Mahoney's complaint as "grandstanding" and said party officials around the country were largely positive about Steele's stewardship of the RNC.
"The criticism has been all Beltway-driven -- consultants, pundits, anyone within a cab ride to a network studio," Nehring said. "Is the punditry translating to the RNC nationally? The answer is overwhelmingly no. Among people who know what it takes to win elections, they are strongly behind Michael Steele."
Nehring pointed to GOP victories in the Virginia and New Jersey governors' races last year and Scott Brown's upset victory in a special election to replace Democratic Sen. Edward M. Kennedy in Massachusetts in January as evidence that the RNC under Steele has been effective and has spent money well.
Some RNC members have criticized Steele, who is black, for suggesting in an interview with ABC's "Good Morning America" that he is held to a higher standard because of his race.
"I've said publicly that I don't think he's getting more scrutiny because he's black," said Dick Wadhams, an RNC member from Colorado.
The Republican National Committee raised $11.4 million in March, nearly $2 million less than the Democratic National Committee, in spite of an electoral environment that appears to favor Republicans.