As Republicans stream into Cleveland to nominate Donald J. Trump for president, they confront a party divided and deeply imperiled by his racially divisive campaign. He has called for cracking down on Muslims and undocumented immigrants, stoked fears of crime and terrorism and repeatedly declared that the United States is in a war for its very survival.
But amid gloom about Republican prospects in November, Mr. Trump may have endangered the party in a more lasting way: by forging a coalition of white voters driven primarily by themes of hard-right nationalism and cultural identity.
Republicans have wrestled for years with the push and pull of seeking to win over new groups of voters while tending to their overwhelmingly white and conservative base. Now, Mr. Trump’s candidacy may force them into making a fateful choice: whether to fully embrace the Trump model and become, effectively, a party of white identity politics, or to pursue a broader political coalition by repudiating Mr. Trump’s ideas — and many of the voters he has gathered behind his campaign.
With his diatribes against Islam, immigration from Mexico and economic competition from Asia, Mr. Trump has amassed dominant support from restive white voters. His political approach would have Republicans court working-class and rural whites, mainly in the South and Midwest, at the grievous cost of alienating minorities and women, who often decide presidential races.
In his choice of running mate, Mr. Trump moved to further shore up his support among Midwestern whites. Passing over a throng of nonwhite Republicans recently elected to high office, he settled on Gov. Mike Pence of Indiana, whose only demonstrated appeal is to conservative-leaning whites in the Rust Belt.
The coalition that carried Mr. Trump to the nomination he will formally claim at the Republican National Convention this week in Cleveland is likely to remain a powerful force on the right, even if he is defeated in November. But its continued sway within the party could suffocate Republicans at the national level, stifling attempts to expand beyond a dwindling base of aggrieved older voters.
A starkly different path forward for Republicans would involve rejecting that base and the ideas that Mr. Trump has used to assemble it.
In order to build a winning party again, some Republican leaders say, the party will have to disavow Mr. Trump’s exclusionary message, even at the price of driving away voters at the core of the Republican base — perhaps a third or more of the party.
This approach would amount to a highly risky lurch away from the faction that made Mr. Trump the Republican nominee, and toward a community of female, Latino and Asian voters who have never been reliable Republicans. Should the effort falter, and Republicans fail to win a second look from these Democratic-leaning groups, they could find themselves stranded with virtually no base at all.
If they are divided over the proper course forward, Republican leaders agree that a wrenching struggle is coming.
House Speaker Paul D. Ryan predicted that the aftermath of the election would bring “a fight for the soul of our party,” and said Republicans would have to reject the politics of racial resentment, which he called “a loser.”
“Our job is not to preach to a shrinking choir; it’s to win converts,” said Mr. Ryan, who has endorsed Mr. Trump but criticizes his pronouncements with regularity.
Senator Jeff Flake, Republican of Arizona, an outspoken critic of Mr. Trump who plans to skip the convention, said more bluntly that the party should be prepared to break with Mr. Trump and the voters who have cheered his pledges to deport millions of undocumented immigrants and ban Muslims from entering the country.
“You’ve got to hope that, if this race keeps going the way it looks to be going, that it’s enough of a jolt to wake people up and say we don’t want to be relegated to second place in every future presidential campaign,” Mr. Flake said.
He suggested a purge of racists from the party that would recall the expulsion of the John Birch Society, a fringe nationalist group, from Republican ranks a half-century ago.
“Those who want a Muslim ban, those who will disparage individuals or groups — yes, we ought to, we need to,” Mr. Flake said.
Many Republicans balk at the idea of executing a kind of mass deportation from within the party’s base, arguing that Mr. Trump has demonstrated the potency of issues outside the establishment Republican catechism, like the mix of trade protectionism, draconian immigration restrictions and resistance to foreign wars summed up in his slogan “America First.”
Republican and Democratic strategists who have studied his coalition believe Mr. Trump’s following may constitute one-third to one-half of Republican primary voters — people drawn principally to his willingness to defy the sensitivities of racial politics and to channel populist anger over immigration and economic change.
In a sense, he has expanded to potentially catastrophic proportions the racial and cultural dilemma that confronted Mitt Romney in 2012. Mr. Romney ran to the right on immigration in the primaries, pledging to clamp down on the Mexican border and push undocumented workers out of the country. He won nearly 60 percent of the white vote against President Obama, but lost by historic margins among Hispanic and Asian voters.
Mr. Trump appears likely to lose nonwhites in an even greater landslide than Mr. Romney.
Whit Ayres, a Republican pollster who wrote a book urging Republicans to pursue nonwhite voters, said Mr. Trump would have to win about 70 percent of whites to make up the difference, an exceedingly daunting political task.
That kind of political calculus has not yet budged Mr. Trump’s most fervent backers, who see the 2016 race as a battle over national identity.
The appeal of a Trump-like message may go beyond even the share of primary voters that Mr. Trump captured: Exit polls found solid majorities of Republican primary voters supportive of his pledge to block Muslims from entering the country. In the general election, polls show most voters oppose that plan.
Last fall, the immigration reform group FWD.us conducted polls in three swing states testing arguments against Mr. Trump, and found that most voters opposed his pledge to round up and deport millions of undocumented immigrants — but “very conservative” Republicans tended to support the idea.
Mr. Trump’s approach is an alluring path to prominence on the right: Already, a handful of up-and-coming Republicans from the party’s conservative wing have moved to court his core voters. Some have argued his message could be more potent in the hands of a less flawed messenger.
“Sometimes being tough-minded is the compassionate approach,” Mr. Cotton said, rejecting the less-strident “compassionate conservatism” espoused by George W. Bush. “I don’t see much compassion in continuing to bring a million legal immigrants to this country a year when our work force participation rate is at historic lows, when we have record high numbers of people on food stamps and disability.”
Mr. Flake invoked the Republican National Committee’s so-called autopsy report after the 2012 campaign, which argued for minority outreach and immigration reform, as a sign of the futility of the party’s predicament.