In a written statement released by the White House, Mr. Bush said his action came at Miers' request.
A White House spokesman denied the president was giving in to pressure from conservatives who vehemently opposed Miers' nomination, but Democrats accused Mr. Bush of bowing to the "radical right wing of the Republican Party."
In a letter to Mr. Bush, Miers said, "I am concerned that the confirmation process presents a burden for the White House and our staff that is not in the best interests of the country."
Mr. Bush said he understands and shares Miers' concern. He blamed her withdrawal on calls in the Senate for the release of internal White House documents that the administration has insisted were protected by executive privilege.
Read Harriet Miers' letter to the president (.pdf)
"It is clear that senators would not be satisfied until they gained access to internal documents concerning advice provided during her tenure at the White House disclosures that would undermine a president's ability to receive candid counsel," Mr. Bush said.
"Harriet Miers' decision demonstrates her deep respect for this essential aspect of the constitutional separation of powers and confirms my deep respect and admiration for her."
Mr. Bush said that despite her withdrawal, Miers would remain as White House counsel. He pledged to name a new nominee "in a timely manner."
Senate Majority Leader Bill Frist said he expected a replacement within days and wants to hold hearings by Christmas. Equally likely was that retiring Justice Sandra Day O'Connor would remain on the court until early next year while her replacement is sought, a prospect that concerns many conservatives.
Miers' surprise withdrawal stunned Washington on a day when the capital was awaiting news on another front – the possible indictment of senior White House aides in the .
CBS News chief White House correspondent John Roberts reports Miers telephoned the president Wednesday night at 8:30 p.m. to say she wanted to withdraw. She delivered her resignation letter to him shortly after 8:30 a.m. Thursday in the Oval Office.
In her letter to Mr. Bush, Miers noted that members of the Senate had indicated their intention to seek documents about her service in the White House in order to judge whether to support her nomination to the Supreme Court.
"I have been informed repeatedly that in lieu of records, I would be expected to testify about my service in the White House to demonstrate my experience and judicial philosophy," she wrote.
"While I believe that my lengthy career provides sufficient evidence for consideration of my nomination, I am convinced the efforts to obtain Executive Branch materials and information will continue."
The Republican chairman of the Senate Judiciary Committee dismissed the White House line that the Miers withdrawal was somehow linked to protecting privileged documents, reports CBS News correspondent Bob Fuss.
"The committee studiously avoided asking what advice Ms. Miers gave to the president and that limitation would have been continued in any hearing," said Sen. Arlen Specter, R-Pa.