In a statement released by the Arizona Republican Party, Mr. Pearce said the comments were written by someone else and that he had “failed to attribute them to the author.” He said that he was stepping down to avoid a distraction before the November elections, when Arizona voters will elect a new governor.

“This was a mistake,” he said. “This mistake has been taken by the media and the left and used to hurt our Republican candidates.”

A week after the radio show, the Arizona Democratic Party stepped forward to question why there had been no outcry by Republicans about Mr. Pearce’s remarks. “For the first vice chair of the Arizona Republican Party to advocate for forced sterilization is unacceptable,” said D. J. Quinlan, executive director of the Arizona Democratic Party, in a statement issued Saturday.

The silence of Republican leaders, he added, “is even worse.”

“It indicates that they have made a cynical calculation that Russell Pearce and his brand of politics appeals to the most extreme elements of their electoral base,” he continued.

On Sunday, Republican candidates stepped forward one after another to denounce Mr. Pearce. Doug Ducey, Arizona’s state treasurer and Republican candidate for governor, said on Twitter: “I couldn’t disagree more with Russell Pearce’s deplorable comments. They have no place in our discourse.” Mark Brnovich, a Republican running for attorney general, released a statement saying the comments were “cruel” and “unrepresentative of the Republican Party I know.”