Appearing at a news conference on identity theft, Gonzales said he will remain "as long as I can continue to serve effectively."
"There are a series of priorities, a series of objectives, that I want to see accomplished," he said, "and we are working as hard as we can to achieve those objectives. Obviously, as head of an agency ... we worry about questions about morale. The way I do it is by speaking directly to U.S. attorneys. ... "
"I have an obligation to work with the Congress and I will continue to work with the Congress," he said.
"We're going to correct the mistakes that have been made. I have accepted responsibility," Gonzales said. "I've already indicated that I've made mistakes, and I accept responsibility for that."
Earlier, Mr. Bush gave his longtime friend and associate a strong endorsement. Appearing with Gen. David Petraeus at the White House, the president told reporters, "This is an honest, honorable man, in whom I have confidence."
Although the attorney general's Senate testimony last week left even some Republicans calling for him to step down, Mr. Bush said his confidence in Gonzales actually increased, CBS News White House correspondent Mark Knoller reports.
"As the investigation, the hearings went forward, it was clear that the attorney general broke no law, did no wrongdoing," Mr. Bush said.
But the controversy isn't over, reports CBS News chief White House correspondent Jim Axelrod. One key Republican senator tells CBS News that if Gonzales loses his job, it will be after an internal Justice Department investigation is concluded in the next month.
Gonzales has been under fire for what the White House acknowledges was his poor handling of the firing of eight federal prosecutors. He claimed dozens of times at last week's hearing that he couldn't recall key details about the prosecutors' firings or about a key November meeting that documents show he attended.
Mr. Bush said that while some senators did not like the way Gonzales answered the questions, he continues to back his attorney general.
However, key GOP lawmakers continued to raise doubts about Gonzales.
Sen. Arlen Specter of Pennsylvania, the top Republican on the committee, said Monday that keeping Gonzales as attorney general will be "harmful to the Justice Department because he has lost his credibility."
"When he said that he wasn't involved in discussions or deliberations, and then is contradicted by his three top aides and also by documentary evidence, ... his credibility has been substantially undermined," Specter said in Harrisburg, Pa. "And I think it does hurt the administration, and inevitably it hurts the (Republican) party."
Specter added: "As long as (Gonzales is) the attorney general, I will continue to deal with him, but whatever he has to say I will take with more than a grain of salt."
"All of America saw why so many of us had felt for so long that he shouldn't be attorney general," said Sen. Charles Schumer, D-N.Y., a Judiciary committee member and one of Gonzales' most vocal critics. "He was not in command of the facts. He contradicted himself. And he doesn't really appreciate the role of attorney general."
Schumer maintained that Gonzales ought to step down as soon as possible.
Asked whether Gonzales should resign, Senate Judiciary Committee Chairman Patrick Leahy, D-Vt., said: "I don't think he can be effective" if he remains in office.
Mr. Bush spoke about Gonzales during an Oval Office meeting with Gen. David Petraeus, commander of the Iraq war.
"The attorney general went up and gave a very candid assessment, and answered every question he could possibly answer — honestly answer — in a way that increased my confidence in his ability to do the job," Bush said.
White House deputy press secretary Dana Perino said that, "Now there are some people, granted, that don't support the attorney general. But he has the full confidence of the president. We reserve the right to respectfully disagree with friends of ours on Capitol Hill in this regard."
Acknowledging Gonzales' lack of support in Congress, Perino said the Justice Department has "a huge amount of responsibility outside of any dealings with Capitol Hill."
"I think that it was good to get the hearing over with," she said. "People can take a step back and then either ask follow-up questions or move on."