Mr Hancock said it was "vital" for those wishing to express their anger over the US killing to stick to social distancing rules and limit any protests to no more than six people "for the safety of your loved ones".
The appeal came as Mr Hancock announced new rules requiring visitors and outpatients at NHS hospitals to wear face coverings and staff to wear surgical masks at all times from 15 June.
Asked whether he wanted police to break up any gatherings of seven or more people taking part in Black Lives Matter demonstrations over the weekend, the health secretary replied: "That is very much an operational matter for the police."
Protest events have been announced in London's Trafalgar Square, Manchester, Leicester, Sheffield and Ipswich for Saturday and at a number of cities including Edinburgh and Glasgow and outside the US Embassy on Sunday.
The planned demonstrations are expected to attract large crowds, after thousands of people - including Star Wars actor John Boyega - marched from Hyde Park to Parliament on Wednesday, most of them breaching the social distancing requirement to keep two metres apart at all times.
But Mr Hancock told the daily Downing Street coronavirus briefing: "Like so many I am appalled by the death of George Floyd and I understand why people are deeply upset but we are still facing a health crisis and coronavirus remains a real threat.
“The reason that it is vital that people stick to the rules this weekend is to protect themselves and their family from this horrific disease.
“So please for the safety of your loved ones do not attend large gatherings, including demonstrations of more than six people.”
Setting out the new rules on face coverings in hospitals, Mr Hancock - for the first time hosting a daily briefing without scientific or medical advisers - said: “As the NHS reopens right across the country, it’s critically important to stop the spread amongst staff, patients and visitors too.
“So today we’re setting out that all hospital visitors and outpatients will need to wear face coverings."
Mr Hancock said that people in hospitals - and particularly those working there, whether in a clinical setting or not - were more likely to catch coronavirus than anywhere else.
He said: "To offer even greater protection we’re also providing new guidance for NHS staff in England which will come into force again on June 15. All hospital staff will be required to wear type one or two surgical masks.
“This will cover all staff working in hospital, it will apply at all times – not just when they are doing life-saving work on the frontline – and it will apply in all areas, except those areas designated as Covid-secure workplaces.”
Mr Hancock told the daily briefing the Government was upgrading the guidance to ensure that “even as the virus comes under control” hospitals are a place of “care and of safety”.
“We’ve also strengthened infection control in care homes and we’re working with the social care sector on how this approach can apply appropriately in social care too.
“It’s about protecting the NHS and social care, which means protecting our colleagues who work in the NHS and in social care.”
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