"We have increasing confidence that vaccinations are effective against all variants including the Indian variant," Johnson said in parliament.
"We have one of the strongest border regimes anywhere in the world," he added, after a day of confusion from ministers on Tuesday about whether travel is allowed to "amber" countries including most of Europe.
Travel to amber countries should only be undertaken for "any emergency or extreme reason" such as family illness, the prime minister said.
Health secretary Matt Hancock meanwhile is facing calls to explain his claim that the rise of the Indian variant in Britain is due to a refusal by some people to get vaccinated.
'World-first' booster study
Official data has shown the variant actually took hold due to travellers coming from India, raising more questions about the government's delay in adding the country to the red list when Pakistan and Bangladesh were already on it.
The data shows that positive tests among travellers coming from India were higher than those from Bangladesh and comparable to Pakistan, well before the new restrictions took effect for India on 23 April.
Hancock told parliament on Wednesday that 2,967 cases of the B1617.2 variant have now been identified in Britain, up nearly one-third since Monday.
Announcing "surge testing" in several more areas, he said "the race between the virus and the vaccine has got a whole lot closer".
But so far the government has insisted it remains on track to lift virtually all restrictions on public life from 21 June, after a successful vaccination campaign.
On Wednesday Britain recorded another 2,696 cases of Covid-19 and three more deaths, taking the total to 127,694, one of the world's worst tolls.
The health secretary also announced that Britain is launching a "world first" study into administering a booster shot, based on trials of seven coronavirus vaccines currently approved in Western countries.
Nearly 3,000 participants will take part in the trial from early June and findings are expected in September to inform policymaking on whether a third shot is needed for the winter months, when respiratory infections spike.
Hancock further announced he will host an in-person meeting of G7 health ministers on 3-4 June at Oxford University, where the AstraZeneca vaccine was developed, to discuss ways of preventing future pandemics.