NEW YORK — Democratic leaders nationwide sought to exact a political price from Republican officials and candidates Monday for continuing to support Donald Trump after his explosive remarks challenging the objectivity of judges with Mexican or Muslim backgrounds.

In an unusually coordinated series of attacks leveled from congressional offices and the Senate floor, in state Capitols and sidewalk protests, Democrats excoriated Trump as racist and demanded that Republicans either stand behind his comments or condemn him and even rescind endorsements of his candidacy.

Democrats received unexpected ammunition from Trump himself, who, in a conference call with allies Monday, urged them to defend his criticisms of a federal judge’s Mexican heritage — and then rebuked his campaign staff for having suggested otherwise.

Trump’s doggedness, and his chastisement of his own aides, contributed to a sense of powerlessness among Republicans who said they increasingly saw no way to influence Trump’s behavior or to convince him that his actions could hurt the party in competitive House, Senate, and governor’s races.

Reince Priebus, chairman of the Republican National Committee, has held discussions with Trump about his attacks on Judge Gonzalo P. Curiel of US District Court, who is overseeing a suit against the now-shuttered Trump University, according to a Republican briefed on the talks. But Priebus has had similar conversations over many months with Trump, to little avail. And other senior Republicans said there was confusion about whether it was worth approaching any of Trump’s aides about doing an intervention with him.

Republican candidates in tough races this fall were left to fend for themselves. Senator Kelly Ayotte of New Hampshire called on Trump to retract his comments about Curiel, calling them “offensive and wrong.” Other candidates laid low. Senator Marco Rubio of Florida, a former rival of Trump’s, and Senator Susan Collins of Maine both criticized Trump, yet neither they nor Ayotte indicated they would reject his candidacy.

“I continue to hope that Mr. Trump will rethink his position and take back those words and show respect for the separation of powers doctrine that is enshrined in our Constitution,” Collins said in an interview.

No prominent elected Republican came to Trump’s defense unreservedly. And others found themselves wondering aloud what it would take for Republicans who have endorsed him to start jumping ship.

Senator Lindsey Graham of South Carolina, another former primary rival of Trump’s, urged Republicans who have backed Trump to rescind their endorsements, citing the remarks about Curiel and Trump’s expression of doubt Sunday that a Muslim judge could remain neutral in the same lawsuit, given Trump’s proposed ban on Muslim noncitizens entering the country.

“This is the most un-American thing from a politician since Joe McCarthy,” Graham said.

“If anybody was looking for an off-ramp, this is probably it,” Graham added. “There’ll come a time when the love of country will trump hatred of Hillary.”

No other modern presidential campaign has unfolded like this, and gleeful Democrats have concluded that one of their best strategies for the general election is to hold Republicans accountable for each new Trump bombshell.

In Indiana, state Democratic Party officials held a news conference Monday to assail Trump’s remarks about Curiel, who was born in the state and graduated from Indiana University’s law school. But the party’s chairman, John Zody, spent even more time criticizing Governor Mike Pence, a Republican seeking reelection this year, and Republican House and Senate candidates in Indiana for being “completely silent on the fact that Donald Trump is questioning the integrity of a Hoosier.”

Only a few Republican officials in the state have criticized Trump’s comments as inappropriate.