During a COVID-19 briefing in Downing Street, the prime minister warned people not to think that "fresh air in itself automatically provides some immunity".
While being outdoors is "crucial for health, physical and mental well-being", he said the government's advice must be taken seriously.
"You have to stay two metres apart and you have to follow the social distancing advice," he said.
He added: "Take this advice seriously, follow it, because it's absolutely crucial."
If people do not do that, there is "no doubt that we will have to bring forward further measures", Mr Johnson said.
The prime minister has warned that Britain is only "two or three" weeks behind Italy, where the number of dead is now more than 5,400, making it the worst outbreak anywhere in the world.
Mr Johnson told reporters that the "best bet" to stop the spread of coronavirus was to "stay at home if you possibly can".
He also thanked the "vast majority" of people who are "behaving incredibly responsibly".
The Communities department said the government was "not telling parks to close", adding that it was for "local councils to decide whether or not to keep their parks open".
People who do not have coronavirus symptoms can go for a walk as long as they are keeping away from each other, it said.
There have been warnings that some major parks in London could be shut if social distancing advice is not followed amid large numbers of visitors.
London is seeing about a third of all UK deaths from COVID-19.
The Royal Parks, which manages a number of open spaces in the capital, has urged people to "keep a safe distance from others".
It added on Twitter that if people "do not follow social distancing guidelines, we will have no choice but to close the parks".
Police on Sunday prevented traffic from entering Richmond, Bushy and Greenwich parks, with only workers holding a permit allowed to use the parking areas.
Hammersmith and Fulham Council has already shut its parks "until further notice", while Kew Gardens has been closed "as a precaution to limit the spread of coronavirus".
London mayor Sadiq Khan tweeted a stark message to all those in the capital, telling them that "life has changed".
"Do not leave your home unless you have to," he said.