In a press conference with Singapore's prime minister, Obama added that Trump has shown he is not up to the task, in light of his handle on foreign affairs and his comments on military families.
"The notion that he would attack a Gold Star family that made such extraordinary sacrifices on behalf of our country, the fact that he doesn’t appear to have basic knowledge around critical issues in Europe, in the Middle East, in Asia means that he is woefully unprepared to do this job," the president said at the White House.
Trump has come under fire in recent days for controversial comments he made about the family of fallen Army Capt. Humayun Khan, who was killed by a car bomb in Iraq in 2004. Khan's father, Khizr, took on the Republican presidential nominee in a speech at the Democratic National Convention last week, criticizing him for his anti-Muslim rhetoric and suggesting Trump has "sacrificed nothing and no one."
In an interview Sunday with ABC News' George Stephanopoulos, Trump criticized Khizr Khan's wife for standing silently next to her husband at the convention and tried to defend himself against Khan's accusations.
"I work very, very hard. I've created thousands and thousands of jobs, tens of thousands of jobs, built great structures. I've had tremendous success. I think I've done a lot," Trump said.
Trump later tweeted, "Mr. Khan, who does not know me, viciously attacked me from the stage of the DNC and is now all over T.V. doing the same - Nice!"
Republicans leaders, such as Senate Majority Leader Mitch McConnell and House Speaker Paul Ryan, have since praised the Khan family and other Gold Star families for their sacrifices, but they have not withdrawn their support for Trump.
The president said Republican denunciations of Trump "ring hollow" as leaders in the GOP continue to endorse him.
"There has to come a point at which you say, 'Enough,'" Obama said.
"The question that I think that they have to ask themselves is, if you are repeatedly having to say in very strong terms that what he has said is unacceptable, why are you still endorsing him? What does this say about your party that this is your standard bearer?" Obama said. "This isn’t a situation where you have an episodic gaffe. This is daily and weekly where they are distancing themselves from statements he's making."