Boris Johnson has urged the country to “work peacefully, lawfully” to defeat racism and discrimination.
Writing in The Voice, the PM said the government could not ignore the anger and “undeniable feeling of injustice” sparked by George Floyd’s killing.
But he said the cause was at risk of being “undermined” by a minority of those attacking police and property.
The UK, he said, had made “huge strides” in tackling racism in recent decades but more had to be done.
Meanwhile, the cabinet is due to meet on Tuesday to discuss the next steps for easing the coronavirus lockdown in England.
The death of George Floyd, 46, in Minneapolis, in the United States, after a white police officer was filmed kneeling on his neck, triggered an international outcry and sparked days of mass protests in cities across the UK.
While the demonstrations have been largely peaceful, they turned violent in London on Saturday when police officers were confronted with flares, and a statue of wartime leader Winston Churchill was vandalised.
In an article for the Voice, the prime minister said Mr Floyd’s death had “awakened an anger and a widespread and incontrovertible, undeniable feeling of injustice, a feeling that people from black and minority ethnic groups do face discrimination: in education, in employment, in the application of the criminal law”.
“We simply cannot ignore the depth of emotion that has been triggered by that spectacle, of a black man losing his life at the hands of the police,” he wrote.
“We who lead and who govern simply can’t ignore those feelings because in too many cases, I am afraid, they will be founded on a cold reality.”
‘More to do’
While he believed the UK was a much less racist society than it was 40 years ago, the PM said he “heard” the Black Lives Matter protesters and accepted much more needed to be done to ensure everyone was treated equally.
“We must also frankly acknowledge that there is so much more to do – in eradicating prejudice, and creating opportunity, and the government I lead is committed to that effort.”
Thanking those who abided by social distancing while taking to the streets, the PM warned that further mass demonstrations endangered the UK’s efforts to bring the coronavirus epidemic under control.
— BBC News