Carr also wrote to two US broadcasters, PBS and NPR, to find out whether the program was aired in the US and had broken FCC regulations, the Telegraph reported.
The UK public broadcaster apologised for the speech edit, which it said gave the "mistaken impression that President Trump had made a direct call for violent action" on the day of the US Capitol attack on January 6, 2021.
The BBC said the splicing of the speech was an "error of judgment" but refused to pay financial compensation, after the US president's lawyers threatened to sue for $US1 billion in damages unless a retraction and apology were published.
In an episode broadcast in June 2022, Newsnight reportedly played an edit of his speech which was similar to the one used in the Panorama program.
Carr's letter reportedly outlined how the two segments of speech that had been spliced together had been spoken almost an hour apart.
"In doing so, the BBC program depicts President Trump voicing a sentence that, in fact, he never uttered. That would appear to meet the very definition of publishing a materially false and damaging statement," he said.
"As you may know, broadcasters regulated by the FCC have a legal obligation to operate in the public interest. Those public interest requirements include prohibitions on news distortion and broadcast hoax."
In January, Carr asked the FCC Enforcement Bureau to open an investigation of PBS and NPR and said "Congress is actively considering whether to stop requiring taxpayers to subsidise NPR and PBS programming".
In May, Trump signed an executive order aiming to slash public subsidies to the broadcasters over "bias" in their reporting.
BBC chairman Samir Shah will face questions from UK MPs next week.
The edit was brought to public attention after a leaked report from former editorial standards adviser Michael Prescott, which raised concerns about the Panorama programme Trump: A Second Chance?.
The program was broadcast a week before the US election in November 2024.
Two of the BBC's most senior executives, director-general Tim Davie and news chief Deborah Turness, have quit.
Trump has a history of suing news organisations in the US and is currently engaged in legal action with the New York Times and the Wall Street Journal.
A BBC spokesperson said: "We have had no further contact from President Trump's lawyers at this point. Our position remains the same."
