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.