White House hopeful Donald Trump lost no time Sunday in claiming a massacre in a gay nightclub showed he has been right about Islamic extremism.

The Republican flag-bearer demanded that President Barack Obama stand down if he refuses to blame the attack on what Trump called "radical Islam".

And the property mogul went to Twitter to double down on his promise to ban Muslims from travelling to the United States if elected.

Obama had earlier branded the attack an act of "terror" and "hate" but the FBI investigation into the slain suspect's motives is continuing.

The attack by a US citizen left 50 dead in the worst mass shooting in US history and has been claimed by a media source close to the Islamic State group.

In a series of Tweets starting just hours after the shooting began and as reports were still sketchy, Trump said the tragedy supported his views.

"Appreciate the congrats for being right on radical Islamic terrorism," he said. "I don't want congrats, I want toughness and vigilance. We must be smart!"

Later, he followed up with a strongly-worded statement.

"In his remarks today, President Obama disgracefully refused to even say the words 'Radical Islam'. For that reason alone, he should step down," he said.

"If Hillary Clinton, after this attack, still cannot say the two words 'Radical Islam' she should get out of this race for the presidency."

Clinton, Trump's Democratic rival for the White House, had been due to campaign with Obama on Wednesday, but cancelled the event while the FBI probe continues.

"If we do not get tough and smart real fast, we are not going to have a country anymore," Trump warned.

"Because our leaders are weak, I said this was going to happen -- and it is only going to get worse.

"I am trying to save lives and prevent the next terrorist attack. We can't afford to be politically correct anymore," he warned.

Trump has previously argued that events like the San Bernardino shooting in December last year show that the United States should ban Muslim travelers.

In his statement, Trump did not repeat this call for a visa ban -- promising simply to make a detailed national security speech on Monday.

But he quickly returned to Twitter to repeat and underline what has proved one of his most controversial campaign promises.

"What has happened in Orlando is just the beginning," he tweeted. "I called it and asked for the ban. Must be tough."