Trump wrote that the protester, Martin Gugino, could be a member of the anti-fascist movement antifa, which the president and other Republicans have blamed for violence at the protests.
Trump offered no evidence other than a report by pro-Trump channel One America News Network, which also cited no evidence.
"Buffalo protester shoved by Police could be an ANTIFA provocateur. 75 year old Martin Gugino was pushed away after appearing to scan police communications in order to black out the equipment. @OANN I watched, he fell harder than was pushed. Was aiming scanner. Could be a set up?" Trump tweeted.
Two Buffalo police officers have been arraigned on felony assault charges after a viral video showed one of them, Officer Aaron Torgalski (39), shoving Gugino and the other, Officer Robert McCabe (32) walking past his still body after he fell to the sidewalk.
As Gugino tumbles, his head strikes the pavement and trickles blood, according to the video.
Trump's post appeared just hours before the funeral in Houston of Floyd, an African American who died on May 25 this year after a white police officer in Minneapolis pinned him with a knee to the neck for nearly nine minutes.
New York Governor Andrew Cuomo said he was "disgusted" by Trump's tweet and urged the president to apologise.
"He accused this man of being associated with antifa - no proof whatsoever. No facts, just an assertion," Cuomo said at his daily news briefing. "I mean, if there was ever a reprehensible, dumb comment, and from the president of the United States," said Cuomo, a Democrat.
"At this moment of anguish and anger what does he do? Pours gasoline on the fire....He should apologize for that tweet."
Erie County District Attorney Erie County District Attorney John Flynn, who is prosecuting the officers, declined to comment on Trump's tweet about Gugino, who remains hospitalized.
"This case is going to get tried in court before someone in a black robe and I have no further comment," said McCabe's lawyer Tom Burton.
Lawyers for Torgalski and Gugino did not immediately respond to a Reuters request for comment.
