Boris Johnson has been accused of “dog-whistle” Islamophobia by a former Conservative chair after he compared Muslim women in burqas to “letterboxes” and “bank robbers.”

Sayeeda Warsi said the lack of action by the party over Johnson’s comments showed it was “business as usual.”

Labour demanded that the Conservative chair, Brandon Lewis, refer the former foreign secretary for mandatory equalities training, amid renewed calls for the party to open an independent investigation into Islamophobia in the party.

In a letter to Lewis, the shadow equalities minister, Naz Shah, said Johnson’s comments were “ugly and naked Islamophobia” and said Lewis should abide by his word to give Tory members diversity training to combat Islamophobia.

The party was also criticised by the Finsbury Park mosque imam, who was praised for his heroism during the terror attack last year. He said the Tory party was “in denial” over the extent of Islamophobia.

Johnson said he did not agree with a ban on the face veil, but compared Muslim women in burqas to bank robbers and rebellious teenagers. In his column for the Telegraph, he said he would expect his constituents to remove face coverings when talking to him at his MP’s surgery, comments which Shah said fell foul of equalities law.

“As a Muslim woman, I am appalled that this kind of ugly and naked Islamophobia has been published in a national newspaper and so far appears to be tolerated by your party leadership,” Shah wrote in the letter that was also sent to the women and equalities minister, Penny Mordaunt.

Shah said Johnson’s suggestion that he should be “fully entitled” to ask a constituent who came to him for advice or assistance to remove her veil was “grossly insulting and Islamophobic... it would also potentially be unlawful if he were to carry it out”.

The MP said the Conservative party had vowed to tackle Islamophobia, pointing to a ConservativeHome article where Lewis said he would set up diversity training. “If no action is taken against Mr Johnson – for example, at the very least requiring him to attend a course of training and engagement with the Islamic community – what faith can Muslim people have in your public statements?” she wrote.

Lady Warsi, who has previously called for an inquiry into Islamophobia in the Conservative party after it was raised by the Muslim Council of Britain (MCB), tweeted that the comments were “dog-whistle” from Johnson. The MCB said Johnson was “pandering to the far-right” and underlined the need for a deeper investigation by the Conservative party.

Sayeeda Warsi (@SayeedaWarsi) “Tories are in denial on Islamophobia- says the hero praised around the world for his brave & compassionate response to the murderous attack in #FinsburyPark” @standardnews

Meanwhile @Conservatives its business as usual with #Boris #Burkha dog whistle 😔https://t.co/KXFl3GtgEo

The Finsbury Park imam, Mohammed Mahmoud, said there was a general lack of engagement with the Muslim community from ministers. Writing for the Evening Standard on the anniversary of the van attack that killed one worshipper and injured 12 others, he said: “Despite the rising scale and severity of Islamophobia and anti-Muslim hatred, the response from the government has been lacklustre, or worse, in denial.”

Other Labour MPs condemned Johnson. David Lammy called him a “pound-shop Donald Trump” and accused him of “fanning the flames of Islamophobia” for political advantage. Jess Phillips said she would report Johnson to the Equality and Human Rights Commission.

In his column, Johnson said schools and universities should be entitled to tell students to remove a veil if a student “turns up … looking like a bank robber”.

“It is absolutely ridiculous that people should choose to go around looking like letter boxes; and I thoroughly dislike any attempt by any – invariably male – government to encourage such demonstrations of ‘modesty’,” he wrote. Businesses and government agencies should be able to “enforce a dress code that enables their employees to interact with customers”, including by allowing them to see their faces, Johnson said.

Denmark introduced a burqa ban last week, with fines of around 1,000 kroner, following similar moves in France, Austria and Belgium.

Johnson said he did not support a blanket ban on wearing a face veil in the UK. “You risk turning people into martyrs, and you risk a general crackdown on any public symbols of religious affiliation, and you may simply make the problem worse,” he wrote.

The Conservative party did not respond to a request for comment.