He said it was possible that Mr Obama "did sometimes" but "other presidents did not call".
In the latest installment of 'What Despicable Thing Did Donald Trump Say Today?', the president made the disgusting claim at a press conference on Monday that, unlike Barack Obama and other former presidents, Trump calls the family members of fallen USA soldiers to pay his respects.
The comment came in response to a question from a reporter who asked why the president hadn't spoken publicly regarding the deadly October 4 attack in Niger that left four Green Berets dead following an ambush.
"I don't know if he did", he said, trying to reverse his earlier comment.
In an email to TPM, a former Obama White House official strongly disputed Trump's claim.
Trump has so many flaws, and there are so many reasons he's ill-suited to the presidency and ill-equipped for its responsibilities, but sometimes, I simply wish he weren't such a small man in a big office.
He and Ms Obama also had a Gold Star Christmas tree every year, honouring the families of fallen soldiers. "But when we got on Marine One to fly back to the White House, the president was looking out the window, and then he looked at me and he said, 'That mama sure was mad at me.' And then he looked out the window and he said, 'And I don't blame her a bit.' And a tear rolled down his cheek, but he didn't wipe it away, and then we flew back to the White House".
In an impromptu and free-wheeling press conference on Monday, the US President was asked about the four US soldiers killed in Niger last week.
A White House official said Kelly did not receive a call from Obama at the time. "Now, as far as other representatives, I don't know". "I like, when I can, the combination of a call and also a letter".
However, Kelly and his wife did attend a breakfast for Gold Star families at the White House in 2011 and sat at the table of former First Lady Michelle Obama, according to ABC News.
"I think that the inattention to this issue is not acceptable", Reed told CNN.
The specifics regarding the Kellys aside, Trump's attack on Obama for supposedly disrespecting a Gold Star family is rich, given his own attacks on a Gold Star family during the presidential campaign.
"I will, at some point during the period of time, call the parents and the families - because I have done that, traditionally", he added. "I can tell you, my policy is I have called every one of them". He said he was also told by staff of other occasions when Obama contacted families.
Gen Kelly has yet to comment on the president's latest remarks.