It comes after the former foreign secretary faced condemnations from across the political divide for the inflammatory remarks he made in a column on Monday.
In the most high-profile intervention so far, Mr Lewis said he agreed with one of his Tory colleagues that Mr Johnson’s comments were “offensive”.
But Labour said an apology was “not good enough”, and called on Theresa May to condemn the newspaper column “unequivocally”.
Earlier, Alistair Burt, a foreign office minister, told the BBC Radio 4’s Today programme: “I would never have made such a comment, I think there is a degree of offence in that, absolutely right.
“What he was trying to make a serious point about is the UK Government will not enforce any clothing restriction on anyone.”
Writing in the Daily Telegraph, the former foreign secretary described the burka as “ridiculous” and “weird” and said women wearing them looked like letter boxes or bank robbers. His remarks came after Denmark decided to impose fines on those wearing the religious headgear in the streets.
Lady Warsi accused Mr Johnson of adopting the “dog whistle” tactics of former Donald Trump aide Steve Bannon in the hope of attracting support from right-wing Tories for an eventual leadership bid.
Repeating her call for an independent inquiry into Islamophobia in the Conservative Party, Lady Warsi told Channel 4 News: “Muslim women should not be a useful political battleground for Old Etonians.
“It is crass and it must stop, and it must be condemned by the leadership right from the Prime Minister down.”
Labour’s shadow women and equalities minister Naz Shah added: “An apology isn't good enough. Boris Johnson's comments weren't accidental, they were a calculated attack in a national newspaper, made weeks after he reportedly met with Steve Bannon.
“I suggested to Brandon Lewis yesterday that the former Foreign Secretary needs to attend training and engagement with the Muslim community for Muslims to have any faith that the Conservative Party is taking this issue seriously. Clearly the Tory party has an issue with Islamophobia, but over 24 hours later the Prime Minister is still yet to say a word.
“Theresa May must condemn Boris Johnson's comments unequivocally and order an inquiry into Islamophobia in her party.”
Downing Street has so far avoided criticising the former foreign secretary for his newspaper column, with a spokesperson for the prime minister on Monday saying only: “The longstanding government position on this is clear – we do not support a ban on wearing of the veil in public.
“Such a prescriptive approach would be out of keeping with British values such as religious intolerance and gender equality.”
But the Conservative MP Andrew Bridgen said Mr Johnson did not need to apologise. He told BBC Radio 4's The World At One programme the former foreign secretary had been trying to raise the subject in a “lighthearted way”.
“I think if you can get your point across with a little bit of humour it's very much appreciated by the public,” he added.
“Boris is seen as a clear challenger for the leadership in due course and it's interesting the characters, Alistair Burt, love him to bits, and the party chairman, and we all know which side they are batting on.”
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