WASHINGTON (CNN) -- As the White House turns its attention to finding a new Supreme Court nominee, conservative activists relieved at Harriet Miers' withdrawal are vowing to oppose President Bush's next nominee unless the candidate has solid conservative credentials."I think [conservative groups] will swing into action again" if they disagree with his next pick, said Phyllis Schlafly, president of Eagle Forum. "The judicial issue was a major issue in the 2004 elections, and it was a reason why many people voted for Bush even though they might have been unhappy [with him] for other reasons."No announcement concerning a new nominee was expected Friday.Schlafly said Bush has "a dozen" possible people that conservative activists would accept, while she singled out U.S. Attorney General Alberto Gonzales as a nominee they could not embrace. (Possible nominees)"It is hard to find anyone else like Miers, unless you are talking about Gonzales," she said. "Most of the same arguments would be there. I think [conservative critics] would simply pick up where they left off."Leaders on the far right opposed the nomination of Miers, the White House counsel, because they did not believe she would fit in the conservative mold of current Justices Antonin Scalia and Clarence Thomas. And for many of these leaders, the same fear lies with Gonzales.Choosing Gonzales, who preceded Miers as White House counsel, also could conflict with Bush's explanation for Miers' withdrawal -- concerns that senators wanted documents of privileged discussions between the president and his top lawyer. (Full story)A senior administration official told CNN that the next choice will be based at least in part on the "lessons learned" from Miers' nomination. (See video on lessons learned -- 1:53)Meanwhile, Americans surveyed in a national poll Thursday expressed mixed feelings about Miers' withdrawal as a nominee for the Supreme Court. (Full story)In addition to Miers' perceived lack of conservative credentials, the White House also could consider two other points of criticism on her nomination: her lack of experience as a judge or with constitutional law; and her close ties to the president, which prompted Democratic concerns about her judicial independence.Conservatives expressed concern that Miers' lack of a paper trail left few clues on how she might rule on issues important to them. (Watch video on why the nomination fell apart -- 2:51)By comparison, Bush's choice of John Roberts, a judge on the U.S. Court of Appeals for the D.C. Circuit, to replace the late Chief Justice William Rehnquist encountered little resistance, winning Senate approval in a 78-22 vote. Bush said he would name his new selection "in a timely manner," and after the selection process that resulted in the Roberts and Miers nominations, considerable research has been done.Democrats, however, urged the president to have patience with the process and to pick a nominee in the judicial "mainstream.""The president now should take his time," said Sen. Charles Schumer. "The president should do it right: slowly, deliberately, carefully, with real consultation and real consensus.""These are very difficult times for the country, and the nation cries out for unity," said the New York Democrat, who serves on the Judiciary Committee. Addressing his comments to Bush, Schumer added, "Please help bring America together with a choice that unites, not divides us."Vermont Sen. Patrick Leahy, another Democrat on the committee, called for "the necessary independence from partisan factions."Conservatives said they would refrain from endorsing a particular candidate before Bush chooses a nominee to replace retiring Justice Sandra Day O'Connor, who was nominated by President Reagan and became a moderate, swing vote on the high court.But Tony Perkins, president of the Family Research Council, said leaders within the conservative movement would not remain silent on what type of person they want on the bench."Very clearly, there is going to be a call for a nominee that can have the enthusiastic support from the people that supported the president," he said.Gary Bauer, president of the conservative American Values organization, acknowledged that "there is a great relief" among his conservative allies that Miers ended her nomination bid. He said it offers Bush a second chance to nominate someone with a defined conservative philosophy."It has been a difficult three weeks, not only for her and the president but for those of us on the outside who were not able to justify or rationalize the nomination," Bauer said.Senate Majority Leader Bill Frist of Tennessee called for "a highly qualified nominee who is committed to upholding the Constitution and who believes in the limited role of a judge to interpret the law and not legislate from the bench." (Watch video on potential nominees -- 3:42)Choosing a woman?While conservative activists seem to agree the next candidate for the court should subscribe to a conservative philosophy, there might be a split on whether the next nominee should be a woman -- an option supported by O'Connor and first lady Laura Bush.Traditional Values Coalition Chairman Louis P. Sheldon said he favors replacing O'Connor with another woman and cited three appellate court judges as possible choices: Edith Hollan Jones and Priscilla Owen of the 5th U.S. Circuit Court of Appeals and Janice Rogers Brown, who recently joined the U.S. Court of Appeals for the District of Columbia Circuit."I think we should have a woman this time," Sheldon said. "Isn't the [justice] retiring a woman?"Another judge on the 5th Circuit, Edith Clement, was among the finalists interviewed by Bush before he selected Roberts.But a senior official with Concerned Women for America said gender should not be a determining factor as Bush decides who should be his next nominee."For us it is not about sex, race or creed," said Lanier Swann, the conservative group's director of government relations. "It is really about their ability to fairly interpret the Constitution."One potential female candidate whom some observers have called a "Roberts clone" is Maureen Mahoney, a private attorney at Latham & Watkins in Washington. Like Roberts, she once clerked for Rehnquist.The only woman on the Judiciary Committee, Sen. Dianne Feinstein, objected to the mention of Brown as a candidate. "That would be a real problem," the California Democrat said. "Her opinions are extraordinarily, extraordinarily out of the box. They didn't even make sense." She called for the president to name a nominee "in the mainstream of American jurisprudence, who can help bring this nation together." Both Brown and Owen have been the subject of Democratic filibusters. (Full story)One conservative leader suggested Bush could help bring together a Republican Party in "disarray" by nominating a conservative that Democrats would vehemently oppose."A fight I think would be helpful," said the leader, who spoke on the condition of anonymity. "What will bring people together is to have a common goal, and that would be the nomination of a conservative nominee."But not all conservative activists were pleased that Miers chose not to stay and fight for her nomination."I am just very disappointed that it had to come to this," said Roberta Combs, president of the Christian Coalition of America. "I wish she would have stayed in. I don't think she deserved this."The most recent Supreme Court nominee to withdraw was Douglas Ginsburg in 1987, whom President Reagan selected after Robert Bork was rejected. Ginsburg withdrew his nomination after admitting he had used marijuana, and Reagan's subsequent pick of appeals court judge Anthony Kennedy was successful.CNN's Mark Preston contributed to this report.