Representative Peter T. King of New York said his fellow Republicans cannot be timid about criticizing activists like the Rev. Al Sharpton, who Mr. King said used racially charged terms to portray the killings of African-Americans by the police in Ferguson, Mo., and on Staten Island.

“I just think if anything, Republicans somehow get scared off of the issue if race comes up and they somehow back away, and that allows people who want to make it a racial issue to be heard,” Mr. King said in a telephone interview.

What makes this moment more complex for Republicans, however, is that Mr. Sharpton is not the only one who has criticized police mistreatment of minorities and the broader justice system: Leading Republicans, including Senator Rand Paul of Kentucky, governors and Christian conservatives, have been rethinking issues ranging from the militarization of the police to sentencing guidelines.

Newt Gingrich, a former House speaker, acknowledged that he was “very hard-line” on crime during his political ascent in the 1980s and ’90s. The second legislative proposal in the “Contract With America,” which he and fellow Republicans used to win the House majority in 1994, was termed “The Taking Back Our Streets Act,” and included longer prison sentences, money for prison construction and additional police officers. But Mr. Gingrich argued that changes to the criminal justice system are needed and would not be deterred after the events over the past month.

Image Former Mayor Rudolph W. Giuliani of New York City said on Fox News recently that the protests were leading to violence. Credit... Rob Kim/Getty Images

“It’s not hard to distinguish between anti-police rhetoric and being concerned that, both in terms of human development and financial costs, we adopted some policies that didn’t work,” Mr. Gingrich said.