The raid on 28 January was the first such operation authorised by Mr Trump.
Bill Owens, whose son William "Ryan" Owens was killed, told the Miami Herald that "the government owes my son an investigation".
"Why at this time did there have to be this stupid mission when it wasn't even barely a week into his administration? Why?" he said in an interview with the newspaper published on Sunday.
"For two years prior, there were no boots on the ground in Yemen - everything was missiles and drones - because there was not a target worth one American life. Now, all of a sudden we had to make this grand display?''
The raid - approved by President Trump just six days after he took office - is believed to have killed several civilians, including children.
Three other Americans were wounded in the raid, planning for which began under the Obama administration.
A report in the New York Times alleged that the mission had been compromised early on - something the commandos knew from intercepted communications, the paper said.
"They kind of knew they were screwed from the beginning," it quoted a former Seal Team 6 official as saying.
But the military disputed this, saying there was no evidence the mission was compromised.
White House spokeswoman Sarah Huckabee Sanders said on Sunday that she believed President Trump would support an investigation.
"The mission has a lot of different critics, but it did yield a substantial amount of very important intel and resources that helped save American lives and other lives,'' she said.
Mr Owens said he was told that his son was killed early on in a fierce gunfight that erupted with AQAP militants.
A flag-draped casket bearing his body was flown to Dover Air Base on 1 February for a private ceremony. Mr Owens said he only learned that President Trump and his daughter Ivanka were on their way when he was already there.
"I told them I didn't want to make a scene about it, but my conscience wouldn't let me talk to him," he said.
A military veteran, he told the Miami Herald that Mr Trump's comments during the presidential campaign about the family of a dead Muslim US soldier had disturbed him.
He said that the White House should not "hide behind" his son's death to ignore criticism of the raid.
William "Ryan" Owens, a 36-year-old married father of three, was a member of the elite Seal Team 6.
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