(NBC News) - When Donald Trump blasted President Obama Sunday for failing to make reference to Islam in connection with the Orlando nightclub massacre, the GOP's presumptive presidential nominee was renewing a longstanding criticism of the White House's carefully calibrated rhetoric about terrorism.

Obama "disgracefully refused to even say the words 'Radical Islam'," Trump said in a statement Sunday. "For that reason alone, he should step down. If Hillary Clinton, after this attack, still cannot say the two words 'Radical Islam' she should get out of this race for the Presidency."

Although the Orlando gunman pledged allegiance to ISIS, Obama didn't mention Islam in his remarks about the mass shooting Sunday. In televised remarks at the White House Monday, Obama said the killer had been influenced by "extremist ideology," but the president did not use the modifier "Islamic."

He did say that "one of the biggest challenges" is the "propaganda and perversion of Islam that you see on the internet."

Obama and his secretary of state, John Kerry, have said that they don't use terms like "Islamic extremism" or "radical Islam" because they believe doing so would grant undeserved religious legitimacy to terrorist movements such as the Islamic State. Citing Islam as a factor risks framing counterterrorism as a war between the West and Islam, they have said.

"They are not religious leaders -- they're terrorists," Obama said in February. "And we are not at war with Islam. We are at war with people who have perverted Islam."

Until recently, presumptive Democratic presidential nominee Hillary Clinton, who was Obama's first secretary of state, shared the president's view on the issue. But in response to Trump on Monday, Clinton pivoted.

"Whether you call it radical jihadism, radical Islamism, I think they mean the same thing," she said on CNN. "I'm happy to say either."

But, she added, "What I won't do, because I think it is dangerous for our efforts to defeat this threat, is to demonize and demagogue and, you know, declare war on an entire religion. That plays right into ISIS' hands."

Trump, in his rhetoric, often conflates Islamic extremism and terrorism with the Muslim religion. In a phone interview with CNN Monday, he said that "for some reason, the Muslim community does not report people like this," referring to Omar Mateen, the Orlando gunman, who may have had mental problems.

In fact, 28 percent of homegrown counter-terrorism prosecutions had their start in tips from family or community members, according to the New America Foundation, which analyzed the data on 505 cases since 2001.

In advocating for his proposal to ban Muslim immigration to the U.S, Trump said on ABC Monday: "We have many people coming in whose hate is equal to his and just as bad and even worse, frankly. And, we have to stop people from coming in. We have no documentation, we don't know where they come from. They could be ISIS, in fact, some of them have cell phones with ISIS flags on them."

Mateen was born in New York and was a U.S. citizen. Syed Rizwan Farook, who carried out the San Bernardino massacre last year with his immigrant wife, was born in Chicago and was a U.S. citizen. The majority of homegrown terrorism since 2001 was carried out by U.S. citizens, according to the New American Foundation.

But even many people who don't share Trump's views on Muslim immigration accuse Obama of failing to acknowledge the connection between a strain of Islamic fundamentalist ideology and terrorism.

"This is not just about words," Tulsi Gabbard, a Hawaii Democrat and Iraq war veteran, told Fox News last year. "It's not about semantics. It's really about having a real, true understanding of who our enemy is and how important that is, that we have to understand what their motivation is and what their ideology is — the radical Islamic ideology that is fueling them."