Mr Hunt questioned the Government's decision not to cancel large gatherings after a stark warning from Prime Minister Boris Johnson that many more families would "lose loved ones before their time".
Asked on BBC Newsnight what he thought about the decision not to cancel large gatherings, Mr Hunt said: "I think it is surprising and concerning that we're not doing any of it at all when we have just four weeks before we get to the stage that Italy is at.
"You would have thought that every single thing we do in that four weeks would be designed to slow the spread of people catching the virus."
He added that he is "personally surprised that we're still allowing external visits to care homes".
Mr Hunt said the UK is in a "national emergency" and that many people "will be surprised and concerned" that the UK is not moving sooner.
He said other countries that have taken tougher action appear to have been successful in turning back the tide of the virus.
It comes as France became one of the latest European countries to close all schools, universities and nurseries, while in the US, all major sport has been suspended and Broadway performances have been stopped for a month.
On Thursday, Ireland announced the closure of all schools and childcare facilities and other public spaces such as museums, while Scotland banned gatherings of more than 500 people.
Mr Johnson has said the latest Government approach is aimed at protecting the elderly and those most vulnerable to the disease, but said precautions would mean severe disruption across the country "for many months".
Between 5,000 and 10,000 people in the UK are thought to be infected with Covid-19 already.
Mr Johnson told reporters at a press conference on Thursday: "We've all got to be clear: this is the worst public health crisis for a generation.
"It is going to spread further and I must level with you, I must level with the British public: many more families are going to lose loved ones before their time."
He said that from now on, people with even mild coronavirus symptoms, including a continuous cough or high temperature, must stay at home for at least seven days.
School trips abroad should be stopped, people over 70 with serious medical conditions are being told not to go on cruises, and officials warned the advice is likely to develop so that entire households could be told to self-isolate.
The Government said it would not move to close schools yet as the evidence for its effectiveness is lacking, though this will be kept under review.
It is also considering banning large events, mostly due to the burden such events place on public services.
High-profile people to be diagnosed with Covid-19 include Arsenal head coach Mikel Arteta and BT boss Philip Jansen.
Chelsea's full squad has also gone into self-isolation after winger Callum Hudson-Odoi tested positive for coronavirus.
Sophie Gregoire Trudeau, wife of Canadian Prime Minister Justin Trudeau, has tested positive for Covid-19 and the couple are in isolation.
Mr Trudeau's office said he is feeling well and will continue to work.
Following the PM's comments, Health Secretary Matt Hancock tweeted: "We have today moved to the next stage in our coronavirus action plan. I know how worrying this time is for so many.
"We will base our decisions on the bedrock of the science - and do the right thing at the right time to keep people safe."
Keith Neal, emeritus professor in the epidemiology of infectious diseases at the University of Nottingham, said the measures would help contain the spread of the virus.
He said: "The plans are sensible, it is very easy to say more needs to be done, but there is little evidence to make any decision."
England's Chief Medical Officer, Professor Chris Whitty, said the peak of the outbreak is most likely still 10 to 14 weeks away.
He said it is feared people will become "fatigued" by more stringent measures if they are brought in too soon and therefore they would lose their maximum effect.
It comes as:- The FTSE 100 closed down by more than 10 oer cent yesterday as fears over Covid-19 sparked the index's worst bloodbath since 1987, while markets fell in Asia today.- The Players Championship golf tournament in Florida was cancelled after one round with organisers saying it was "the right thing to do" because of the virus spread.- The number of people who have tested positive for the virus in the UK rose to 596 while the death toll in British hospitals is 10.- Nepal cancelled the Mount Everest climbing season until at least the end of April.- The Electoral Commission recommended local elections in May be postponed- Stormont ministers insisted it is not the right time to close Northern Ireland schools over coronavirus, despite the announcement by the Irish Republic that schools will be shut south of the border.
One expert said there is a danger that people in whom the virus presents as similar to a common cold may not realise they should self-isolate.
Professor Jonathan Ball, professor of molecular virology at the University of Nottingham, said the Government advice, which specifies persistent cough and raised temperature, "needs careful consideration by those charged with mitigating the impact of this pandemic".