Addressing reporters outside the Red Cross shelter, flanked by his wife Melania, Trump said: “She owes the country an apology.”
Trump was responding to a question from a reporter about Ms Warren’s alleged ancestry.
A recent DNA report suggested the Senator – whom Trump has regularly referred to as “Pocahontas” – does have Native American roots.
Trump said: “What’s the percentage? One 1,000th?
“When you have the percentage, tell me what the percentage is.”
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Trump witnessed first-hand the devastation brought to Florida by Hurricane Michael, as rescuers scoured for survivors, while hundreds of thousands of people suffered electricity blackouts.
Trump and the First Lady Melania gave out water bottles at an aid centre in Lynn Haven, a city with a population of roughly 18,500 people near Panama City in north-western Florida.
The couple had flown in on a helicopter from Elgin Air Force Base about 100 miles to the west.
Trump said: “To see this personally is very tough – total devastation.”
The President later went to Georgia to witness the damage there.
Michael’s impact is estimated to have killed at least 18 people after the calamitous storm hurtled into Panhandle last Wednesday.
It was one of the most powerful storms on record to have ever collided with the continental US.
With maximum sustained winds of 155 miles per hour (250 kph), Michael hit the Florida Panhandle as a Category 4 hurricane on the five-step Saffir-Simpson scale.
Ignatius Carroll, a Miami fire captain who leads a Federal Emergency Management Agency rescue team, said: “The next phase is recovery.
“We start using the dogs for larger rubble piles that were created by the storm.”
Thousands of rescuers, including volunteers, are still scouring remote areas of the Florida Panhandle for the missing.
They include 46 in Mexico Beach, according to ABC News.
The town took a direct hit from the hurricane, with at least one recorded fatality.
Rescue teams went through debris by hand, rather than with machines, so as not to destroy any bodies, Mexico Beach Councillor Linda Albrecht said.
Ms Albrecht said: “We expect to find everybody, because that’s our mentality.
“We expect everything to work out, but who knows what’s down the road?”
The Florida Division of Emergency Management said power was returning in most areas.
But at least 85 percent of customers in four mainly rural Panhandle counties were still without electricity on Monday.
Officials said it could be weeks before power returns to the most-damaged areas.
Port St Joe Mayor Bo Patterson, whose town of 3,500 was without power, said: “We’re living in the daylight, and living in the dark once night gets here.”