The foreign secretary said footage of Floyd’s killing while he was being restrained by police in the US was “very distressing”.
But he added he did not want to comment on what the president had said.
Trump has appeared to support tough tactics being used by the National Guard as protests spread across America in response to the killing.
Hundreds of people have taken to the streets in protest across the US in recent days, with more than 1,300 having been arrested in 16 cities since Thursday night.
Trump has characterised some of the protests as “mob violence” and said he wanted police to “stop it cold”.
Asked about Trump’s response, Raab told the BBC’s Andrew Marr Show on Sunday morning: “I’ve long kept to the self-imposed guidance not to comment on what president Trump says or indeed other world leaders, it is not really what my job is.”
He also told Sky’s Sophy Ridge on Sunday: “I’m not going to start commenting on the commentary or indeed the press statements that other world leaders make, or indeed the US president.
“Footage of what happened to George Floyd was very distressing, as has been the scenes across America of the rioting and some of the violence.
“And what we do know is that the lead suspect has now been charged with murder, there is a federal review and we want to see de-escalation of all of those tensions and American come together.”
Protests over the death of Floyd and other police killings of black people in the US grew on Saturday from New York to Tulsa to Los Angeles.
Police vehicles were set ablaze and injuries mounted as the country convulsed through another night of unrest after months of coronavirus lockdowns.
Trump issued tweets taunting protesters and praising the secret service, who used shields and pepper spray to push back people gathered outside the White House to protest Floyd’s death and the president’s response.
The president tweeted he had watched from inside as officers “let the ‘protesters’ scream & rant as much as they wanted, but whenever someone …. got too frisky or out of line, they would quickly come down on them, hard – didn’t know what hit them”.
The protests – which began in Minneapolis following Floyd’s death after a police officer held a knee to his neck until he stopped breathing – have left parts of that city a grid of broken windows, burned-out buildings and ransacked stores.
The unrest has since become a national phenomenon as protesters decry years of deaths at police hands.
More than 1,300 people have been arrested in 16 cities since Thursday, with more than 500 of those happening in Los Angeles on Friday.