In that same interview, Trump was asked to explain why he called for Obama to resign in light of the shooting and he answered, in part: "He doesn't get it or he gets it better than anybody understands — it's one or the other, and either one is unacceptable."
For months, Trump has slyly suggested that the president is not Christian and has questioned his compassion toward Muslims. Years ago, Trump was a major force in calls for the president to release his birth certificate and prove that he was born in the United States. On the campaign trail, Trump has repeatedly stated as fact conspiracy theories about the president, his rivals and Muslims, often refusing to back down from his assertions even when they are proven to be false.
During an appearance on the "Today" show later Monday morning, Savannah Guthrie pushed Trump to explain what he meant in the earlier interview.
"Well there are a lot of people that think maybe he doesn't want to get it," Trump said. "A lot of people think maybe he doesn't want to know about it. I happen to think that he just doesn't know what he's doing, but there are many people that think maybe he doesn't want to get it. He doesn't want to see what's really happening. And that could be."
Guthrie asked Trump why that would be, and the presumptive Republican presidential nominee responded: "Because Savannah, Savannah, why isn't he addressing the issue? He's not addressing the issue. He's not calling it what it is. This is radical Islamic terrorism. This isn't fighting Germany; this isn't fighting Japan, where they wear uniforms."
Trump's spokeswoman and campaign manager have yet to respond to a request for a fuller explanation of Trump's comments about the president.
White House press secretary Josh Earnest said later Monday that he had not spoken with the president about Trump's criticisms, but he emphasized that "when you are focused on something as big as helping the country respond to the worst mass shooting in the nation's history, it's important not to get distracted by things that are so small."
Obama did not mention Trump during brief remarks to reporters in the Oval Office after meeting with his national security advisers Monday morning.
In December, Trump called for a temporary ban on allowing foreign Muslims into the country as a way to protect against terrorist attacks. Recently, as he faced pressure from fellow Republicans, Trump seemed to back away from the religion-based ban, saying that it was merely a suggestion. But after this weekend's mass shooting, Trump has doubled down on his call for a ban, saying that the United States is allowing in terrorists who could stage even worse attacks.
"That man yesterday was sick with hate," Trump said on Fox. "You have many, many people, thousands of people, already in our country that are sick with hate. And people that are around him, Muslims, know who they are, largely. They know who they are. They have to turn them in. They know who they are. They see them."
During a brief interview on ABC's "Good Morning America," Trump was asked how a ban on foreign Muslims could have prevented the Orlando shooting, as the shooter, identified as Omar Mateen, was born in the United States.
What Donald Trump is doing on the campaign trail
"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," Trump said, adding that gun control measures also could not stop a terrorist attack.
"The problem was that we have a maniac, we have a madman. He could have used a bomb. He could have used other things, just as easily, just as easily," Trump said. "It would have been, probably, even more devastating. So that's not the problem."
Trump has repeatedly taken credit for having been "right" on issues of terror, and on Monday he continued to face scrutiny for a tweet he posted on Sunday that read: "Appreciate the congrats for being right on radical Islamic terrorism, I don't want congrats, I want toughness & vigilance. We must be smart!"
"I'm getting thousands of letters and tweets that I was right about the whole situation," Trump said on Fox News. "I mean, I've been right about a lot of things, frankly.... I was right about many, many things."
Trump is scheduled to give a speech about terrorism in New Hampshire on Monday afternoon, and he declined to provide details on the types of policies he might advocate, other than saying: "I'm calling for strength. I'm calling for intelligence."
Sean Sullivan contributed to this report.