CLEVELAND — Mike Hill, a black Republican state representative in Florida, grew steadily more disheartened as he watched television clips of his party’s overwhelmingly white national convention lecturing African-Americans about the police and race relations.

There was Rudolph W. Giuliani, the former mayor of New York, nearly shouting Monday night that the police only wanted to help people, regardless of race. A sea of white convention delegates, cheering wildly as two black speakers ridiculed the Black Lives Matter movement and unconditionally praised law enforcement officers. And a series of speakers pushing Donald J. Trump’s law-and-order message and arguing, as he has, that the United States had lost its way.

“When a lot of white Republicans get together and bring up race, even telling black people how they should see police and the world, it evokes the worst kind of emotion,” said Mr. Hill, who supports Mr. Trump but decided to skip the convention. “We have so few black Republicans to begin with. Talking about race won’t bring us more.”

For many black Republicans, the party’s convention has veered unexpectedly and unhappily toward lecturing and moralizing on issues of race, an off-putting posture at a time when Mr. Trump is staggeringly unpopular with minority voters. He drew support from zero percent of African-Americans in recent NBC News/Wall Street Journal polls in Ohio and Pennsylvania, and he is struggling badly with Hispanics, partly because of his harsh language about Mexicans and immigrants.