And Speaker Paul D. Ryan, Republican of Wisconsin, posted a video on Twitter backing up Mr. McConnell. Mr. Ryan said that any decision regarding the court vacancy “is going to have to go to the American people.”

While many conservative groups are making the same argument, others are still trying to paint Judge Garland as a reflexive liberal unworthy of a seat on the court. One group, the Judicial Crisis Network, has cited past decisions that it says show his support of “legalizing partial-birth abortion, and unleashing unaccountable bureaucratic agencies,” like the Environmental Protection Agency and the Internal Revenue Service, as well as imposing limits on gun rights.

“The president is presenting him as a wonderful moderate who is a cross between Chief Justice Marshall and Mr. Rogers,” said Carrie Severino, chief counsel at the Judicial Crisis Network, referring to the celebrated Supreme Court justice from the 1800s and the former host of a children’s television show. “We need to have someone out there combating the spin about Judge Garland.”

Attacks on Judge Garland risk provoking a backlash from within the party. In recent days, Republican luminaries such as Kenneth W. Starr, the former federal judge and special prosecutor who led investigations of President Bill Clinton over the Whitewater and Monica Lewinsky scandals, and Michael Chertoff, a former federal judge who served as the Homeland Security secretary under President George W. Bush, have praised Judge Garland. In an interview on Friday, Mr. Chertoff called him “a model judge and someone who has great respect and reverence for the Constitution.”

“I can hardly imagine how one could mount a personal attack against Judge Garland,” he added.

FreedomWorks allowed a reporter rare access to the organization as Mr. Levey and his staff put their plan to stop Mr. Obama’s nominee into action from the organization’s bowling alley-shaped war room, a few blocks from Capitol Hill.