“He is a member of a club or society, very strongly pro-Mexican, which is all fine,” Mr. Trump said. “But I say he’s got bias. I want to build a wall. I’m going to build a wall. I’m doing very well with the Latinos, with the Hispanics, with the Mexicans, I’m doing very well with them, in my opinion.”

Judge Curiel, 62, was born in East Chicago, Ind., to parents who had emigrated from Mexico. He graduated from Indiana University’s law school and worked as an assistant United States attorney in the Southern District of California before being appointed in 2007 to the bench in San Diego by Gov. Arnold Schwarzenegger, a Republican. President Obama nominated Judge Curiel to the federal bench in late 2011, and he was confirmed by the Senate in 2012.

Mr. Trump’s broadside against Judge Curiel was one of the most overtly racial remarks he had made in the presidential race, and it exacerbated the tension between some Republicans and their nominee. White, older, working-class voters make up a large portion of the party’s base, and Republicans need to keep the presidential campaign close in order to hold their majority in the Senate. But Mr. Trump’s remarks have offended wide blocs of voters to whom the party must appeal amid national demographic shifts.

And the critiques have raised concerns about how, as president, Mr. Trump would handle the separation of powers between the executive and judicial branches of government. That issue is sacred to conservatives, who have railed against what they see as an abuse of power by Mr. Obama.