"They're pouring in, and we don't know what we're doing," he added later.
Trump tied the Obama administration's immigration policies to Clinton, who served as secretary of state during Obama's first term.
"The burden is on Hillary Clinton to tell us why she believes that immigration from these dangerous countries should be increased without any effective system ... to screen," Trump said.
The Orlando shooter was a US-born citizen.
Trump's tough rhetoric on national security has endeared him to fearful Americans who believe that domestic terrorism threats have grown under the Obama administration.
Even without comprehensive proposals for upgrading the country's national security apparatus, Trump's support among Republican voters grew after terrorist attacks late last year in Paris and San Bernardino, California.
But critics have blasted Trump for his response to the attack on an Orlando LGBT nightclub, at once the largest mass shooting in American history, a terrorist attack and a hate crime. He initially omitted offers of condolences, instead launching into a full-scale political attack against his Democratic rivals.
"Appreciate the congrats for being right on radical Islamic terrorism, I don't want congrats, I want toughness & vigilance. We must be smart!" he tweeted Sunday, one of his first public statements about the attack.
On Monday, Trump asked those attending his speech to observe a moment of silence for the victims. He also offered his support to members of the LGBT community, speaking compassionately about their struggles.
"We express our deepest sympathies to the victims, the wounded and their families. We mourn as one people for our nation's loss, and pledge our support to any and all who need it," Trump said.
"It's a strike at the heart and soul of who we are as a nation. It's an assault on the ability of free people to live their lives, love who they want, and express their identity," Trump added.
Trump's speech, which did not deliver specific proposals for combating terrorism, focused on recriminations against the Obama administration.
He called Obama's motivations into question Monday, insinuating that the president may sympathize with radical jihadism - a statement that carried echoes of the so-called "birther movement," which circulated conspiracy theories about whether the president was born in the United States.
"Well, there are a lot of people that think maybe he doesn't want to get it," Trump said Monday on NBC. "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."
The most scathing attacks on Trump's temperament after the attacks came from Clinton herself, whose campaign has aggressively sought to brand Trump as unsympathetic and uneducated on foreign policy.
Clinton delivered a lengthy response to the attack Monday during a campaign event in Ohio, pledging to make "identifying and stopping lone wolves" a top priority of her administration if she becomes president. She also called for increased coordination and information sharing between government agencies.
Though Clinton did not mention Trump by name in the speech, she offered clear contrasts to his anti-Muslim rhetoric. She also indirectly criticized his political attacks on President Obama, invoking the political and social response to the Sept. 11 terrorist attacks in 2001 as a model for national healing and leadership.
"Americans from all walks of life rallied together with a sense of common purpose on September the 12th, and in the days and weeks and months that followed. We had each other's backs. ... We did not attack each other, we worked with each other to protect our country and to rebuild our city," Clinton said during the speech. "It is time to get back to the spirit of those days, the spirit of 9/12. Let's make sure we keep looking to the best of our country, to the best within each of us."
In several television interviews Monday, Clinton accused Trump of "demagoguery," adding that he has not offered tangible solutions to terrorism.
"I'm not going to demonize and demagogue and declare war on an entire religion. That's just plain dangerous, and it plays into ISIS's hands," she said on NBC Monday morning, referring to the Islamic State by another name.
Trump issued a direct response Monday during his speech: "She has no clue, in my opinion, what radical Islam is, and she won't speak honestly about it if she does in fact know. She's in total denial."
Perhaps the clearest policy difference between Clinton and Trump rests on the question of gun control and whether stricter requirements would prevent such attacks. Clinton called for gun sale restrictions on people being monitored by the FBI's terrorist watch list.
Trump said earlier in the day that he believed that calls for gun-control legislation in the wake of the Orlando shooting would be a mistake.
"If you had some guns in that club the night that this took place ... you wouldn't have had the tragedy that you had," Trump said on CNN Monday morning.
