WASHINGTON — President Obama angrily denounced Donald Trump on Tuesday for his remarks in the aftermath of the shooting massacre in Orlando, warning that Trump, the presumptive Republican presidential nominee, was peddling a “dangerous’’ mindset that recalled the darkest and most shameful periods in American history.

“We hear language that singles out immigrants and suggests entire religious communities are complicit in violence,’’ Obama said at the Treasury Department, without mentioning Trump by name.

His statement, an extraordinary condemnation by a sitting president of a man who is to be the opposing party’s nominee for the White House, came after Obama met with his national security team on the status of the US effort against the Islamic State, a meeting the president said had been dominated by discussion of the Orlando rampage.

“Where does this stop?’’ Obama said of Trump’s approach, noting that Trump had proposed a ban on admitting Muslims to the United States, and that the Orlando assailant, like perpetrators of previous domestic terrorist attacks in San Bernardino, Calif., and Fort Hood, Texas, was an American citizen.

“Are we going to start treating all Muslim-Americans differently? Are we going to start subjecting them to special surveillance? Are we going to start discriminating against them because of their faith?’’ Obama asked, his voice rising with frustration. “Do Republican officials actually agree with this? Because that’s not the America we want — it doesn’t reflect our democratic ideals. It won’t make us more safe. It will make us less safe.’’

Appearing in Pittsburgh as Obama spoke, Hillary Clinton gave a blistering denunciation of her own. She echoed many of the president’s points and even some of his language, assailing Trump’s temperament, ridiculing his proposals, and arguing forcefully that he had failed to meet the gravity of the moment.

“History will remember what we do in this moment,’’ she told hundreds of supporters at a union hall, asking “responsible Republican leaders’’ to join her in condemning Trump. “What Donald Trump is doing is shameful.’’

Her half-hour speech was a point-by-point rebuttal of Trump’s remarks a day earlier, when he issued a searing broadside implying that all Muslim immigrants posed a threat to American security. The nearly simultaneous condemnations of Trump from the president and the presumptive Democratic nominee had the feel of a coordinated assault, although the White House insisted there had been no preplanning.

Trump, unbowed by the criticism, said Obama was coddling terrorists.

“President Obama claims to know our enemy, and yet he continues to prioritize our enemy over our allies and, for that matter, the American people,’’ he said in a statement on Tuesday. “When I am president, it will always be America first.’’

Members of Trump’s party were themselves critical of his rhetoric and proposals.

Speaker Paul D. Ryan, the highest-ranking elected Republican, said at a news conference Tuesday that Trump’s proposed ban on Muslim immigrants was not in the country’s interests, nor did it reflect the principles of his party.

“There’s a really important distinction that every American needs to keep in mind: This is a war with radical Islam. It’s not a war with Islam,’’ Ryan said. “The vast, vast majority of Muslims in this country and around the world are moderate, they’re peaceful, they’re tolerant, and so they’re among our best allies, among our best resources in this fight against radical Islamic terrorism.’’

Senator Jeff Flake, Republican of Arizona, has been among the most outspoken in his party about withholding his endorsement of Trump.

Flake said in a Twitter post that he was “appreciative’’ Ryan had spoken out.

Senator Mitch McConnell of Kentucky, the majority leader, flatly refused to talk about his party’s presidential nominee on Tuesday, an indication of the precarious position in which Trump has placed Republican elected officials.

Obama bitterly rejected criticism from Trump and other Republicans about his steadfast refusal to use the term “radical Islam’’ to describe the Islamic State.

The president said he would not use the wording because he was unwilling to give the Islamic State the victory of acceptance of its vision that it is the leader of a holy war between Islam and the West.

“If we fall into the trap of painting all Muslims with a broad brush and imply that we are at war with an entire religion, then we are doing the terrorists’ work for them,’’ Obama said.

Obama is scheduled to travel to Orlando on Thursday to visit the surviving victims and the families of those killed in the rampage Sunday morning. He was to have traveled to Wisconsin on Wednesday for his first campaign appearance with Clinton since endorsing her last week, but the event was canceled in light of the shootings.

Still, Tuesday’s one-two punch left little doubt that Obama and Clinton plan to savage Trump on the campaign trail. The president was careful not to cast his criticism in political terms and never mentioned Trump’s name even as he clearly targeted him — at one point referring derisively to “politicians who tweet’’ — and his policy proposals. Instead, Obama spoke ominously of the stakes for the nation’s security, and its very identity, if the ideas that Trump has espoused are widely accepted.

“We’ve gone through moments in our history before where we acted out of fear, and we came to regret it,’’ Obama said. “We’ve seen our government mistreat our fellow citizens, and it has been a shameful part of our history.’’

Clinton, in a striking departure from her speech on Monday, when she refrained from saying Trump’s name and said it was “not a day for politics,’’ took direct aim on Tuesday at his penchant for conspiracy theory. She reminded the crowd that he was “a leader of the birther movement’’ questioning Obama’s birthplace.

After the Orlando attack, she noted, Trump suggested on television that Obama sympathized with Islamic terrorists.

“Just think about that for a second,’’ Clinton said. “Even in a time of divided politics, this is way beyond anything that should be said by someone running for president of the United States.’’

“We don’t need conspiracy theories and pathological self-congratulations,’’ she added. “We need leadership, common sense, and concrete plans.’’

Obama staunchly defended his administration’s approach to countering terrorism, listing gains that the United States has made against the Islamic State in Iraq, Syria, and Libya: killing the group’s top leaders, capturing more of its territory, and whittling away at its financial resources.

He also called on Congress to enact gun restrictions that it has so far resisted, including the resurrection of a ban on assault weapons and a measure that would bar people on “no-fly’’ lists because of suspected terrorist ties from buying a gun.

“Enough talking about being tough on terrorism,’’ Obama said. “Actually be tough on terrorism and stop making it as easy as possible for terrorists to buy assault weapons.’’