CBS News correspondent Thalia Assuras reports that the White House has promised to choose a new nominee quickly, perhaps as early as Friday.
Mr. Bush has offered no hint about his thinking on a new nominee, but he isn't starting from scratch. The president already has vetted and interviewed several candidates. That could point to a handful of federal judges believed to have been finalists when Mr. Bush made his doomed choice of Harriet Miers.
Some top contenders are believed to be federal appellate judges Samuel Alito, J. Michael Luttig, J. Harvie Wilkinson, Alice Batchelder, Priscilla Owen and Karen Williams as well as Michigan Supreme Court Justice Maura Corrigan.
CBS News White House correspondent John Roberts reports that Miers, as White House counsel, will help lead the search for her replacement, and the White House has not ruled out going outside the usual list of suspects.
"I don't know that it has to be a judge,"
Mr. Bush could turn to a current or past senator, such as Republican John Cornyn of Texas, believing the Senate would be more likely to embrace one of its own.
But as CBS News correspondent Thalia Assuras reports, some analysts say Mr. Bush still needs to choose a woman or a minority. Names most often mentioned include Priscilla Owen and Janice Rogers Brown, recently confirmed to different benches, but only after a bitter Senate battle.
Outgoing Justice Sandra Day O'Connor, who will have to wait a little longer to hang up her robe, made clear she would like to see another woman succeed her. She told one reporter, "Women constitute one half of the population in this country. It is critical for women to serve in all branches of government."
Miers, a Texas lawyer and loyal Bush confidante who remains White House counsel, was criticized for having thin credentials on constitutional law and no proven record as a judicial conservative.
CBS News correspondent Lee Cowan reports that Miers' friends in Dallas said she deserved better treatment.
"I think she may have suffered and it's very unfair," said Linda Eads. "She would have made a terrific Supreme Court justice."
Read Harriet Miers' letter to the president (.pdf)
The criticism of Miers' credentials lessens the chance that Mr. Bush will pick someone else who has never rapped a gavel on a judicial bench. It also makes it less likely that the president will choose anyone who could be tagged a Bush crony, according to lawyers in regular touch with White House officials involved in the selection process.
Mr. Bush now finds himself in the awkward spot of finding a replacement for a candidate he had described as the most qualified person in the country. He also had said Miers brought a fresh approach because she had never been a judge.
Jan Crawford Greenburg, who covers the Supreme Court for the Chicago Tribune said on the CBS Early Show that as a result of Miers' experience, White House advisers suggested to President Bush that he turn to an experienced federal appeals court judge with a clearly defined judicial philosophy.
Greenburg also says the White House does not feel as compelled to nominate a woman or a minority as it did when it originally began thinking about this nomination.
"This is a difficult moment for the White House because any nominee that would make the conservatives happy, be the pick they hoped for so long is going to be quite upsetting to the Democrats. This is going to be a fight."
Deeply disappointed by Miers' demise, Mr. Bush will be looking for a repeat of the Senate's 78-22 vote to confirm John Roberts as the 17th chief justice.
"I think that the president is likely to return to the short list of people that everyone left to right agreed are the most qualified," said Brad Berenson, a former staff member of the White House counsel's office in the Bush administration. "The lesson of the Roberts and Miers nominations taken together is that there's considerable safety and power in selecting people whose qualifications can't be questioned."
The White House said it was not the firestorm of opposition from Mr. Bush's right flank, but the Senate's demand for documents covered by attorney-client and executive privileges that forced Miers' withdrawal.
Both these reasons makes it unlikely Mr. Bush will choose Attorney General Alberto Gonzales, another Bush ally who has been on the candidate list.
As with Miers, senators would seek documents Gonzales handled when he was White House counsel, and the White House again would claim executive privilege in denying their release. Nominating Gonzales also would re-ignite the very opposition Mr. Bush is trying to dampen on his right from members of his own party who don't think Gonzales is a reliable conservative vote on abortion and affirmative action.