Johnson's comments followed Denmark introducing a ban on burkas in public places.
The PM also said: "These are issues that are openly discussed but in having that discussion it's important that people think carefully about the language that they use, certain terms in describing people's appearance that Boris Johnson used have clearly caused offence, that's why I have said I agree with Brandon Lewis".
But a source close to Mr Johnson said people should "not fall into the trap of shutting down debate on hard issues".
"When Muslim women are being attacked in the street, his comments are risky and can not be laughed off", she said.
The assistant secretary general of the Muslim Council of Britain, Miqdaad Versi, said Mr Johnson was "pandering to the far right". Some suspected Johnson's burqa comments were meant to boost his appeal among right-wing members of the party.
He said: "Boris isn't stupid, this wasn't an off-the-cuff comment after a glass of wine, this was an article that he sat down and wrote for The Daily Telegraph".More news: Del Potro, Fognini enter Los Cabos Open final
"An apology is not enough, she needs to order an independent inquiry into Islamophobia in her party, as requested by the Muslim community, and take action against him".
Mr Johnson admitted he had received a "kick in the pants" from Mr Howard and promised to go to the city in a "spirit of complete humility".
In his column for the Telegraph on Monday, Johnson said Muslim women wearing burqas looked like bank robbers and that schools and universities should be entitled to tell students to remove them.
"I believe women should be able to choose how they dress", Ms May said. "I wish he hadn't accompanied it with a comment that I certainly wouldn't make and I think many people would find offensive, yes".
Johnson declined to comment on Tuesday, but a person familiar with his views said he was speaking up for liberal values and stood by his article. "It's now 24 hours have gone by, the message that members of the Muslim community get is that their concerns are not taken into account".
Conservative Muslim Forum founder Lord Sheikh told The World at One on BBC's Radio 4 that Johnson was "using Muslims as a springboard" as he seeks to lead the party.More news: Ollie Pope plans to replicate Curran´s class in Test arena
The resignation solidified Mr Johnson's position as a leader of the pro-Brexit wing of the Conservative Party, which is deeply divided over its attitude to the EU.
However, Johnson said he did not support a blanket ban the face veil in the UK.
"People are pretty sick to death of a very politically correct Prime Minister who never ever speaks out of turn on any single subject".
A fine of around £120 has already been imposed on a woman wearing a niqab in a shopping centre in the town of Horsholm, after another woman reportedly tried to tear it off.
Boris's notably liberal argument was forgotten, however, as critics conveniently focused on his contention that it was "absolutely ridiculous... to go around looking like letter boxes".More news: Saudi students' year at UPEI in question