"They honour President Obama," he told a rally in Florida on Wednesday. "He is the founder of Isis [Islamic State].
Mr Trump also attacked his Democratic rival for the White House, Hillary Clinton, calling her a "co-founder".
She responded by accusing him of "trash-talking" the US and echoing the talking points of Russian President Vladimir Putin.
Mr Trump stood by his remarks on Thursday, using a sports phrase to say Mr Obama and Mrs Clinton were the Islamic State's "most valuable players".
The Republican presidential nominee has endured 10 days of negative headlines after a string of controversial comments.
Most recently, he appeared to urge his supporters to take up arms against Mrs Clinton to stop her from appointing liberal judges to the US Supreme Court if she wins the election.
The hotel developer-turned-politician denied he was inciting violence, but the daughter of former President Ronald Reagan, who was shot in 1981, condemned his "verbal violence".
Patti Davis wrote on Facebook: "It was heard by the person sitting alone in a room, locked in his own dark fantasies, who sees unbridled violence as a way to make his mark in the world, and is just looking for ideas.
"Yes, Mr. Trump, words matter. But then you know that, which makes this all even more horrifying."
He has often accused the president of allowing the militants to flourish and has regularly suggested he is actually a sympathiser of the group.
How was Islamic State founded?
IS can trace its roots back to the late Abu Musab al-Zarqawi, a Jordanian who formed al-Qaeda in Iraq (AQI) after the US-led invasion in 2003. It became a major force in the insurgency, carrying out dozens of attacks a month.
After Zarqawi's death in 2006, AQI created a militant umbrella organisation, Islamic State in Iraq (ISI). Over the next four years, it was steadily weakened by a US troop surge and Sunni Arab tribal fighters who rejected its brutality.
Abu Bakr al-Baghdadi became leader in 2010 and began rebuilding ISI. The following year, ISI joined the rebellion in Syria, which offered it a safe haven and easy access to weapons, some of them supplied by US allies opposed to President Bashar al-Assad.
The group also exploited withdrawal of US troops from Iraq at the end of 2011 and widespread Sunni anger at the sectarian policies of the country's Shia-led government.
ISI changed its name to Islamic State in Iraq and the Levant (Isis or Isil) in 2013 and began seizing territory in Syria. In 2014, Isis overran large swathes of northern and western Iraq, proclaimed the creation of a "caliphate", and became Islamic State.
The White House has not commented on the "IS founder" claim but a spokesman for Mrs Clinton said: "This is another example of Donald Trump trash-talking the United States.
"What's remarkable about Trump's comments is that once again, he's echoing the talking points of Putin and our adversaries to attack American leaders and American interests, while failing to offer any serious plans to confront terrorism or make this country more secure."
In recent weeks, several leading Republicans have deserted Mr Trump over his outspoken attacks.
Most recently, Senator Susan Collins said she would not be voting for him, pointing to a time he seemed to mock a disabled journalist.
Time Magazine on Thursday reported that the chairman of the Republican National Committee, Reince Priebus, had threatened to withdraw funding from the Trump campaign, and instead direct it to Congressional campaigns.
Mr Trump denies that this conversation ever took place.
Polls suggest support for the embattled candidate has been falling in key battleground states in recent weeks.