Mr Floyd, who was unarmed and in handcuffs, died on 25 May after a policeman knelt on his neck for nearly nine minutes. The officer involved was charged with murder and three colleagues stand accused of aiding and abetting.
The death triggered protests against racial discrimination across the US and world. On Friday, Minneapolis banned the police neck restraint seen in the video of Mr Floyd's death, and California pledged to follow suit.
What did Trump say?
Speaking in the White House Rose Garden on Friday, Mr Trump said: "Equal justice under the law must mean that every American receives equal treatment in every encounter with law enforcement regardless of race, colour, gender or creed.
"They have to receive fair treatment from law enforcement. They have to receive it.
"We all saw what happened last week. We can't let that happen.
"Hopefully George is looking down and saying this is a great thing that's happening for our country. A great day for him. It's a great day for everybody."
Mr Trump, a Republican, added: "This is a great, great day in terms of equality."
The president's critics said he was crassly suggesting Mr Floyd would be posthumously celebrating positive jobs figures.
Mr Trump's defenders said the context of his comments make clear he was referring to his call for equal treatment of all Americans by police.
More on George Floyd's death
What did Biden say?
The Democratic presidential candidate hit back during a campaign speech in Dover, Delaware.
He said: "George Floyd's last words, 'I can't breathe, I can't breathe' echoed all across this nation and quite frankly around the world.
"For the president to try to put any other words in the mouth of George Floyd I frankly think is despicable."
Mr Biden has been sharpening his attacks on the president as the 3 November White House election looms.
While campaigning in his home state of Delaware on Thursday night, he said Mr Trump brings out the "worst" of some Americans and "there are probably anywhere from 10 to 15 percent of the people out there, they're just not very good people".
What do the job figures show?
The latest US monthly jobs report on Friday showed employers added 2.5 million jobs, defying economists' predictions of further decline.
For black workers, however, the unemployment rate crept upwards from 16.7% to 16.8%.
The losses from the coronavirus pandemic have hit minority and low-wage workers hardest.
The Hispanic unemployment rate declined in April from 18.5% to 17.2%. And it fell among white workers to 12.4% from 13.8%.
Who's banning police neck restraints?
The Minneapolis City Council and the Minnesota Department of Human Rights on Friday agreed to ban police neck restraints and chokeholds.
The new policy, which will be enforceable in court, requires any officer regardless of rank to verbally and physically intervene if they witness a colleague using such unauthorised force.
California's Democratic Governor Gavin Newsom said on Friday he would move to end state police training in the use of the "carotid restraint".
San Diego Police Department banned the technique this week in response to the nationwide uproar over events in Minneapolis.
Officials in New York, San Francisco, Chicago and Los Angeles have all previously taken steps to ban or limit the use of chokeholds by members of their police departments.
US protests timeline
25 May 2020
George Floyd dies after being arrested by police outside a shop in Minneapolis, Minnesota. Footage shows a white officer, Derek Chauvin, kneeling on Mr Floyd’s neck for several minutes while he is pinned to the floor. Mr Floyd is heard repeatedly saying "I can’t breathe". He is pronounced dead later in hospital.
26 May
Four officers involved in the arrest of George Floyd are fired. Protests begin as the video of the arrest is shared widely on social media. Hundreds of demonstrators take to the streets of Minneapolis and vandalise police cars and the police station with graffiti.
27 May
Protests spread to other cities including Memphis and Los Angeles. In some places, like Portland, Oregon, protesters lie in the road, chanting "I can’t breathe". Demonstrators again gather around the police station in Minneapolis where the officers involved in George Floyd’s arrest were based and set fire to it. The building is evacuated and police retreat.
28 May
President Trump blames the violence on a lack of leadership in Minneapolis and threatens to send in the National Guard in a tweet. He follows it up in a second tweet with a warning "when the looting starts, the shooting starts". The second tweet is hidden by Twitter for "glorifying violence".
29 May
A CNN reporter, Omar Jimenez, is arrested while covering the Minneapolis protest. Mr Jimenez was reporting live when police officers handcuffed him. A few minutes later several of his colleagues are also arrested. They are all later released once they are confirmed to be members of the media.
Derek Chauvin charged with murder
Former Minneapolis police officer Derek Chauvin, 44, is charged with murder and manslaughter. The charges carry a combined maximum 35-year sentence.
31 May
Violence spreads across the US on the sixth night of protests. A total of at least five people are reported killed in protests from Indianapolis to Chicago. More than 75 cities have seen protests. At least 4,400 people have been arrested. Curfews are imposed across the US to try to stem the unrest.
1 June
President Trump threatens to send in the military to quell growing civil unrest. He says if cities and states fail to control the protests and "defend their residents" he will deploy the army and "quickly solve the problem for them". Mr Trump poses in front of a damaged church shortly after police used tear gas to disperse peaceful protesters nearby.
2 June
Tens of thousands of protesters again take to the streets. One of the biggest protests is in George Floyd’s hometown of Houston, Texas. Many defy curfews in several cities, but the demonstrations are largely peaceful.
4 June
A memorial service for George Floyd is held in Minneapolis. Those gathered in tribute stand in silence for eight minutes, 46 seconds, the amount of time Mr Floyd is alleged to have been on the ground under arrest. Hundreds attended the service, which heard a eulogy from civil rights activist Rev Al Sharpton.
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