The prime minister said the measures are "under constant review" after reports of people breaking the rules by socialising in parks and using supermarkets without wearing face coverings.
Asked whether the government is considering more strict rules, Mr Johnson said "if we feel that things are not being properly observed then we may have to do more".
"Where we have to tighten [the rules], we will", the prime minister said, adding: "Now is the moment for maximum vigilance, maximum observance of the rules. "
Critics of restrictions have said the government should put England under the same rules as the first lockdown, attacking rules that allow nurseries, places of worship, estate agents and public parks to remain open.
The prime minister hopes vaccines will allow England to lift some restrictions by mid-February, if the 13.9 million people in the most vulnerable groups can be immunised.
Mr Johnson said about two million people have so far been vaccinated with around 2.4 million jabs across the UK.
On a visit to a vaccination centre at Ashton Gate stadium in Bristol, one of seven across England to open on Monday, he told reporters: "As I speak to you today, we've done about two million people, maybe a bit more.
"We're at about 2.4 million jabs all in across the whole of the UK."
He warned that vaccinations are "a race against time" but said with the scale of the threat on the NHS, which he said includes a shortage of oxygen in some places.
The aim of vaccinating the 15 million people in the top four priority groups by mid-February is a "massively stretching" but "achievable target".
He warned the public against “false complacency” because the vaccine is being administered, said rules need to be enforced in supermarkets and that people must “avoid mingling too much” after getting takeaway drinks.
The PM told reporters: “I think what (Professor) Chris Whitty had to say this morning was absolutely right. This is a very perilous moment because everybody can sense that the vaccine is coming in and they can see that the UK is vaccinating large numbers of those that need it most.
“My worry is, and Chris’s worry is, that this is the moment when that degree of false confidence, false complacency, and that when you look at what has happened in the NHS that complacency is not merited.
“More important than us just pushing out new rules, people have got to follow the guidance.
“In supermarkets, people need to be keeping their differences, making sure that they’re wearing masks, doing the right thing.
“We need to enforce the rules in supermarkets. When people are getting takeaway drinks, in cafes, then they need to avoid spreading the disease there, avoid mingling too much.”