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Rebuttals to State of the Union addresses are usually directed at the president from a rising member of the opposing party. But Gov. Nikki Haley of South Carolina, a Republican, had a different target in mind on Tuesday night: Donald J. Trump.

“Some people think that you have to be the loudest voice in the room to make a difference,” Ms. Haley said. “That is just not true. Often, the best thing we can do is turn down the volume.”

Ms. Haley acknowledged on Wednesday morning that she was referring to Mr. Trump when she warned Americans not to follow the angriest voices in politics. “Mr. Trump has definitely contributed to what I think is just irresponsible talk,” she told the “Today” show on NBC.

The remarks drew praise from many Republicans and Democrats and even some talk that Ms. Haley, the daughter of immigrants from India, would be a strong pick as a vice-presidential candidate. Ms. Haley said during the NBC interview that she hadn’t thought about any of the vice-presidential rumors, but added: “If a candidate wanted to sit down and talk, I would sit down and talk. That’s a big decision.”

Frank Luntz, the Republican pollster, noted that Democrats liked the idea that a leading Republican was blaming her own party for the hostile tone of politics.

Democrats love @NikkiHaley accepting Repubs' share of responsibility in current state of hostile politics. #SOTU pic.twitter.com/VrzdSwfwM9 — Frank Luntz (@FrankLuntz) January 13, 2016

Denis McDonough, the White House chief of staff, also expressed approval for Ms. Haley at a breakfast in Washington on Wednesday.

“I have a lot of admiration for the governor,” Mr. McDonough said, adding that some of her work over the last year has been “remarkable,” including her response to the church shooting in Charleston, S.C., and “her very brave and admirable role” in the flap that ensued over taking down the Confederate flag in her state.

Still, Mr. McDonough said he disagreed with many of the decisions that Ms. Haley had made as governor, most prominently her refusal to expand Medicaid in her state as part of the Affordable Care Act.

“By no means am I trying to endorse everything that she’s doing, but I do think that a lot of this, including parts of the speech last night, were admirable,” he said.

Josh Earnest, President Obama’s press secretary, said that the White House took note of Ms. Haley’s remarks and said it was courageous for her to speak out the way she did.

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“She was willing to do something that a lot of other Republicans, leading Republicans, have been unwilling to do, which is to actually articulate a commitment to some core American values that some leading Republican presidential candidates are speaking out against,” Mr. Earnest said.

The reviews were more mixed among Republicans, highlighting the divide within a party whose support is split between hard-line candidates such as Mr. Trump and Senator Ted Cruz of Texas and more moderate figures such as Jeb Bush.

Describing her speech as “remarkable,” Mr. Bush said on Wednesday, “I think she talked about a more broader hopeful, optimistic Republican message, a conservative message that draws people, the great diversity of our country, toward our cause.”

Senator Marco Rubio of Florida also said he was “impressed” with Ms. Haley and, in a moment of self-deprecation, praised her for making it through the remarks without taking a sip of water. Mr. Rubio’s 2013 rebuttal was most notable for his awkward swig.

Carly Fiorina was less impressed, arguing that Americans have a right to be angry about issues such as illegal immigration.

“Look, I don’t speak for Nikki Haley and she doesn’t speak for me,” Ms. Fiorina said on Mike Gallagher’s radio program. “I think you’re correct that it was the wrong note.”

Mr. Trump also made clear that he did not agree with Ms. Haley on Wednesday, calling her “weak” on immigration in an interview on “Fox & Friends” and noting that she had asked him for campaign contributions in the past.

“I feel very strong about illegal immigration,” Mr. Trump said. “She doesn’t.”

Supporters of Mr. Trump were angered that Ms. Haley called him out and many took to Twitter and mocked her Indian heritage, making fun of her Indian given name. Ann Coulter, the conservative commentator, said that Mr. Trump should deport the South Carolina governor.

Later on Wednesday, Ms. Haley said that she would not be deterred from speaking her mind and said that she considered Mr. Trump a friend. In an interview with CNN, she urged him not to take her comments personally and said that she also had concerns about some of his rivals.

“I have disagreements with other presidential candidates,” Ms. Haley said. “Jeb Bush passed Common Core, Marco Rubio believes in amnesty, which I don’t. There’s lots of things, but I will say tone matters, message matters and responsibility matters.”

Julie Hirschfeld Davis and Peter Baker contributed reporting.