The strong rebuke in the White House East Room came after Trump's criticism of the family of a slain Muslim US soldier, along with comments that displayed apparent confusion related to the Russian incursion into Ukraine.
"The Republican nominee is unfit to serve as president," Obama said at a White House news conference with the Prime Minister of Singapore. "He keeps on proving it."
The Trump campaign responded by going after the Democratic nominee as well as the President.
"Hillary Clinton has proven herself unfit to serve in any government office," a Trump statement said, listing a number of policy concerns. "Obama-Clinton have single-handedly destabilized the Middle East, handed Iraq, Libya and Syria to ISIS, and allowed our personnel to be slaughtered at Benghazi."
Later Trump in an interview with WJLA said of Obama: "He's a terrible president. He'll probably go down as the worst president in the history of our country. He's been a total disaster."
Obama on Tuesday described his feelings about Trump as unprecedented, recalling disagreements with previous GOP presidential nominees Sen. John McCain and Mitt Romney -- but never an outright sense they were unfit to serve.
"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 of critical issues in Europe, the Middle East, in Asia, means that he's woefully unprepared to do this job," Obama said.
Speaking alongside Singapore Prime Minister Lee Hsien Loong in the White House East Room, Obama said there are now weekly episodes in which even Republican party leaders distance themselves from Trump.
"There has to be a point at which you say, 'Enough,' " Obama said.
Obama goes after Trump's party
Obama placed responsibility for Trump's statements squarely on his fellow Republicans, many of whom denounced his statements on the slain soldier's family but didn't withdraw their support.
"What does this say about your party that this is your standard-bearer?" Obama asked of GOP leaders. "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. There has to be a point at which you say, 'This is not somebody I can support for president of the United States, even if he purports to be a member of my party.' "
Obama said that denunciations from Republicans of Trump's remarks "ring hollow" without an accompanying withdrawal of support.
"I don't doubt their sincerity. I don't doubt they were outraged by some of the statements that Mr. Trump and his supporters made about the Khan family," Obama said. "But there has to come a point in which you say, 'Somebody who makes those kinds of statements doesn't have the judgment, the temperament, the understanding to occupy the most powerful position in the world.' "
Trump and the family of the slain soldier have been locked in an increasingly bitter dispute over Muslims in America and the nature of patriotic sacrifice.
After Khizir Khan, who lost his son in a suicide bombing in Iraq, declared at last week's Democratic National Convention that Trump had "sacrificed nothing," the Republican nominee claimed he'd been "viciously attacked" and questioned why Khan's wife, Ghazala, didn't make her own remarks.
Criticism from Trump's own party came swiftly, including in a lengthy statement from McCain, whom Trump previously derided for having been taken captive in the Vietnam War. But he and other top GOP leaders, including Senate Majority Leader Mitch McConnell and House Speaker Paul Ryan, made little indication they would withdraw support for the Republican candidate.
Trump has also taken flak for appearing unaware that Russian forces had annexed Crimea in early 2014, saying on ABC's "This Week" Sunday that President Vladimir Putin is "not going into Ukraine." Later, he argued that the people of Crimea "would rather be with Russia than where they were" -- an argument that Putin himself has made in justifying his annexation of the disputed Ukrainian territory.
Trump looks for better Russia ties
And he's maintained his view that fostering better ties to Putin would benefit areas of American foreign policy, including in the fight against ISIS. "If we could get along with Russia, wouldn't that be a good thing, instead of a bad thing?" Trump asked Monday during a campaign appearance in Pennsylvania.
Trump also took heat for calling on Russia last week to release deleted emails from Clinton if they had obtained them in a hack of the Democratic National Committee, which US officials have fingered Moscow for.
At the news conference, Obama said the alleged Russian hacking wouldn't necessarily prompt a complete freeze in relations between the the country and the United States.
Noting that the FBI was still investigating the hack, Obama said cybersecurity was just another dispute of many between Putin and himself.
"If in fact Russia engaged in this activity, it's just one on a long list of issues that me and Mr. Putin talk about," Obama said. "I don't think that it wildly swings what is a tough, difficult relationship that we have with Russia right now."
Obama's administration, through the Justice Department's national security division, continues to investigate the hack. Neither the White House nor the FBI have publicly blamed Russia for the intrusion. But federal officials said there is strong evidence indicating the breach was perpetrated by hackers working on behalf of Russia intelligence.
Trump has brushed off Democratic charges that Putin could be behind the hack in order to tip the US election in his favor, claiming he's never met the Russian leader and doesn't maintain a relationship with him. In statements from only a few years ago, however, Trump said he enjoyed warm ties with Putin.
Obama's remarks Tuesday intensify his patten of inserting himself into the rancorous presidential race. While the President has freely criticized Trump in the year since the businessman entered the race, his denunciations have come faster and harsher in the last several weeks.
At last week's convention, Obama named Trump repeatedly, arguing the candidate was ignorant of facts and intent on dividing the nation. And in June, Obama lit into Trump's response to the mass shooting in Orlando, saying he was peddling a dangerous vision.
The President is expected to play an outsized role on the campaign trail in the coming months as Clinton works to motivate the supporters who helped elect Obama to office twice.