Reid also said Mr. Bush should pledge not to pardon any aides convicted as a result of the investigation into the disclosure of CIA officer Valerie Plame's identity.
"There has not been an apology to the American people for this obvious problem in the White House," Reid said. He said Mr. Bush and Cheney "should come clean with the American public."
Reid added, "This has gotten way out of hand, and the American people deserve better than this."
Cheney's chief of staff, I. Lewis "Scooter" Libby, resigned Friday after he was indicted on five charges relating to statements he made to the FBI and a grand jury investigating the Plame leak.
Reid also said that Karl Rove, the president' closest political adviser, should step down. Rove has not been charged with a crime.
The closest the indictment comes to Rove is its discussion of an unnamed senior White House official who talked to columnist Robert Novak about Plame and discussed the matter with Libby. That could describe Rove.
The prosecutor in the CIA leak case has said his investigation is "not quite done," but declined comment on Rove during a news conference on Friday.
"If you ask me any name, I'm not going to comment on anyone named, because we either charged someone or we don't talk about them," Special Counsel
Patrick Fitzgerald said.
the Special Counsel's statement (.pdf).
When the investigation began, the White House denied that Rove had been involved. Mr. Bush promised to fire anyone on his staff responsible for such a leak. He later stepped back, saying just that he would remove aides who committed crimes.
"I think Karl Rove should step down," Reid said. "Here is a man who the president said if he was involved, if anyone in the administration was involved, out they would go."
Mr. Bush and Cheney gave glowing endorsements and expressed no criticism of Libby after the senior White House adviser was indicted, resigned and lost his security clearance.
Cheney called Libby "one of the most capable and talented individuals I have ever known." Mr. Bush said Libby "has worked tirelessly on behalf of the American people and sacrificed much in the service to this country."
Reid said he was disappointed that Mr. Bush and Cheney expressed support for Libby in their public statements.
"The vice president issues this very terse statement praising Libby for all the great things he's done. Then we have the president come on camera a few minutes later calling him Scooter and what a great patriot he is," Reid told ABC's "This Week."
Sen. John Cornyn, R-Texas, said it was premature to discuss a presidential pardon because no one has been convicted in the investigation.
"People who actually were trying to use this, of course, to the president's political disadvantage, I think, are going to be disappointed by the fact that this appears to be limited to a single individual," Cornyn said.
Reid said the Libby indictment and other scandals in the Republican-led government, including the indictment of former House Majority Leader Tom DeLay of Texas and an investigation of Majority Leader Bill Frist of Tennessee, as well as 2,000 dead in Iraq and high energy prices have had a negative impact on the outlook of Americans.
"I think they're as disappointed as I am ... almost dejected," Reid said.
The president's overall job approval was at 39 percent in an Associated Press-Ipsos survey conducted in early October. The poll also found that only 28 percent of respondents said the country was headed in the right direction.
Sens. Lindsey Graham, R-S.C., and Charles Schumer, D-N.Y., said the White House should conduct its own investigation of the CIA leak. Graham, however, said allegations of illegal activity appeared to be focused only on Libby.
"I think the likelihood of Karl Rove being indicted in the future is virtually zero," Graham said on CBS' "Face the Nation."
"I think this will be seen in history and in politics as Mr. Libby giving false information, if proven, and it will not be about an effort by the vice president to disclose a CIA operative."
Schumer said the investigation showed Cheney's office was in a campaign to discredit Wilson's wife because of his criticism of the administration's use prewar intelligence about Iraq's weapons of mass destruction and that Rove, despite public statements to the contrary, had discussed Wilson's wife with reporters.
If necessary, Schumer said, the president should take Cheney "to the woodshed."
"The president, again, ought to have some nonpolitical person look into this and see what should be done," Schumer said. "The standard shouldn't just be escaping indictment."