Here are five GOP lawmakers who have taken issue with the ways their party's presidential candidate has reacted to the massacre in Orlando. (Sarah Parnass/The Washington Post)

Top Republicans joined with President Obama and other Democrats Tuesday in sharply condemning Donald Trump’s reaction to the nightclub massacre in Orlando, decrying his anti-Muslim rhetoric and his questioning of Obama’s allegiances as divisive and out of step with America’s values.

Trump — who just a week ago signaled an intent to snap his campaign into a more measured tone for the general election — showed no sign of backing down from his suggestions that Obama was somehow connected to or sympathetic with terrorists, telling the Associated Press that the president “continues to prioritize our enemy” over Americans.

In separate appearances, both Obama and his potential successor, likely Democratic nominee Hillary Clinton, blasted Trump’s proposal to ban foreign Muslims from the United States as dangerous and contrary to the nation’s traditions.

A visibly angry Obama also dismissed Trump’s repeated demands for him to use the term “radical Islam” when speaking about the Orlando shootings and other attacks. “Calling a threat by a different name does not make it go away,” Obama said. “This is a political distraction.”

Clinton described Trump’s response to Orlando as rife with “conspiracy theories” and “pathological self-congratulations.”

Speaking in Manchester, N.H. June 13, Republican presidential candidate Donald Trump vowed to shut down immigration from countries "where there is a proven history of terrorism against the United States." Here are key moments from that speech. (Associated Press/Reuters)

The remarkable bipartisan outcry over Trump’s positions — coming at a moment of national mourning after the deadliest mass shooting in U.S. history — set off a new wave of alarm within the GOP over whether the real estate mogul’s promised pivot to the general election would ever materialize. The rift also highlighted the enduring tensions between establishment figures who want to be more inclusive and the bulk of the party, which backs Trump’s proposed Muslim ban and has rallied around him as the presumptive nominee.

Some of Trump’s most ardent backers defended his response to the Orlando attack, saying drastic measures were needed to keep the nation safe. But most Republicans on Capitol Hill tried to distance themselves from Trump’s comments following the terrorist attack on a gay nightclub in Orlando that killed at least 49 people. Senate Majority Leader Mitch McConnell (Ky.) refused to respond to questions about Trump at his weekly news conference.

House Speaker Paul D. Ryan (Wis.) denounced Trump for trying to rally support for his anti-Muslim policies, while others castigated Trump for the accusations he has lobbed at Obama.

“I do not think a Muslim ban is in our country’s interest,” Ryan told reporters. “I do not think it is reflective of our principles, not just as a party but as a country.” He called for “a security test, not a religious test” for immigrants.

In a speech Monday, Trump had reiterated his calls for such a ban and expanded its potential reach to include any country with “a history” of terrorism against the United States and its allies. He blamed the Orlando attack — which authorities say was carried out by a man born in America to Afghan parents — in part on a system that “allowed his family to come here.”

At a rally Tuesday night in Greensboro, N.C., Trump attacked Obama for criticizing him and defended barring foreign Muslims.

“Once again we’ve seen that political correctness is deadly,” Trump said.

“And just so you understand: I have many Muslim friends,” he added at one point. “There doesn’t seem to be assimilation. We don’t know what’s going on.”

Senate Foreign Relations Committee Chairman Bob Corker (R-Tenn.), who has praised Trump at times for his willingness to shake up politics and recently met with the mogul, expressed serious unease Tuesday with how Trump responded to a national tragedy.

“Traditionally, it is a time when people rally around our country, and it’s obviously not what’s occurred, and it’s very disappointing,” Corker said.

Sen. Lindsey O. Graham (R-S.C.), a leading national security hawk, said he had “run out of adjectives” for Trump. “I don’t think he has the judgment or the temperament, the experience to deal with what we are facing,” said Graham, who does not currently support the mogul.

Graham, like other Republicans, took issue with Trump’s apparent suggestions in Monday interviews that Obama may identify with the radical Muslim terrorists. Obama “either is not tough, not smart, or he’s got something else in mind,” Trump told Fox News.

Trump expanded on that Tuesday, saying in an emailed response to questions from the Associated Press: “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.”

Graham said that Trump “seems to be suggesting that the president is one of ‘them.’ I find that highly offensive. I find that whole line of reasoning way off base. My problems with President Obama are his policy choices.”

Sen. Ron Johnson (R-Wis.), who faces a challenging reelection bid, also called Trump’s insinuations about Obama “offensive.”

Speaking after meeting with his National Security Council, Obama dismissed Trump’s many calls for him to change the way he talks about terrorism.

“That’s the key, they tell us. We can’t get ISIL unless we call them ‘radical Islamists,’ ” Obama said, referring to the Islamic State militant group. “What exactly would using this label accomplish? What exactly would it change? Would it make ISIL less committed to trying to kill Americans? Would it bring in more allies? Is there a military strategy that is served by this? The answer is: none of the above.”

At a campaign event in Pittsburgh, Clinton excoriated Trump and challenged Republicans to repudiate him. Clinton said Trump failed to demonstrate an ability to deliver a “calm, collected and dignified response” to the Orlando attack.

“Instead, yesterday morning, just one day after the massacre, he went on TV and suggested that President Obama is on the side of the terrorists,” Clinton said. “Just think about that. Even in a time of divided politics, this is way beyond anything that should be said by someone running for president.”

Trump has also said Obama should “resign” because of his refusal to utter the words “radical Islamic terrorism.” But one of the mogul’s top backers on Capitol Hill said Trump doesn’t expect that to happen.

“What I think Trump’s saying is: You need to get in the game and start leading, or get out of here,” said Sen. Jeff Sessions (R-Ala.). “That’s just his way of expressing it. And I think people understood that. He doesn’t expect President Obama to resign, but he’s saying you can’t do this job effectively if you don’t understand the nature of the threat we face.”

Sessions said there was no discussion at a 90-minute Senate GOP lunch of Trump specifically; instead it focused on terrorism.

Rep. Adam Kinzinger (R-Ill.), an Air National Guard major and leading House GOP voice on national security issues, broke sharply with Trump.

“I guess I appreciate Mr. Trump’s fieriness in talking about it, but you don’t do it by alienating the very people that we need, and those are moderate Muslims,” he said. “We have to use the folks that frankly are not radicalized, which is the vast majority of Muslims, to win this war.”

Nationally, 64 percent of Republican voters said in a recent Washington Post-ABC News poll that they approve of Trump’s Muslim ban — as did 45 percent of independents — while 26 percent of Democrats said they approve.

Last week, Trump delivered a subdued speech that celebrated his primary wins and looked ahead to a matchup with Clinton. His campaign told allies that Trump was strategizing for a new phase of the campaign.

But by this week — after a series of fiery rallies in which he called out enemies by name and then his response to Orlando — many Republicans were left scratching their heads.

Lanhee Chen, a GOP foreign policy expert who served as policy director on Mitt Romney’s 2012 campaign, called Trump’s Monday speech a “huge wasted opportunity.”

“What he has said overall about foreign policy is very troubling,” said Chen, who said he has many issues with the mogul but does not consider himself part of the “Never Trump” wing of the GOP.

Chen said Trump needs to “start defining what his presidency would look like” in “more than just a few sound bites.” But he added: “I’m not holding my breath.”

David Nakamura and Paul Kane in Washington, Abby Phillip in Pittsburgh and Jenna Johnson in Greensboro, N.C., contributed to this report.