The UK’s human rights watchdog is facing demands to launch an inquiry into Islamophobia in the Conservative party.

It comes on the day on which the Equality and Human Rights Commission (EHRC) began a formal investigation into antisemitism in the Labour party.

The Muslim Council of Britain (MCB) filed a complaint of more than 20 pages to the commission on Tuesday, saying calls for the party to allow an independent inquiry had so far been largely ignored. It follows the government’s refusal to adopt a proposed definition of Islamophobia.

The party’s former chair Sayeeda Warsi has described the problem as “institutional”, with a steady stream of accusations against self-professed Tory members coming to light.

Harun Rashid Khan, the secretary general of the MCB, said: “It is a sad day for us to have brought this complaint to the doors of the EHRC, but the concerns of Muslims at large about Islamophobia within the Conservative party have fallen on deaf ears.

“We have taken this step after an unprecedented number of cases have been brought to our attention, suggesting a culture within the Conservative party where Islamophobia is not only widespread, but institutional. We now request the EHRC to look at all the evidence and investigate this matter with great urgency.”

The complaint has five main strands, including Islamophobia among MPs, hostility towards Conservative members, the scale of hostility particularly on social media, the failure of the party’s complaints process and the denial of prejudice by the party’s officers.

The government was accused of disregarding the “safety and security of British Muslims” after rejecting a definition of Islamophobia put forward following a parliamentary inquiry.

The communities secretary, James Brokenshire, pledged to “arrive swiftly at a collective position” earlier this month after senior police officers raised concerns about the potential impact on counter-terrorism powers.

Dozens of members have been suspended for posting Islamophobic comments on social media, including many directed at the home secretary, Sajid Javid, who is standing for the Conservative leadership.

Incidents have included demands by a party member for the government’s only female Muslim minister, Nusrat Ghani, to halt a “Muslim invasion” of the UK. The member was suspended.

Last month, the Guardian revealed two Conservative local election candidates and a woman with an MBE were among 40 self-professed Tory members who had shared or endorsed racist and inflammatory Facebook posts including Islamophobic material.

The Conservative party faced condemnation after reinstating some councillors who had been suspended over racist or prejudiced social media posts.​

An EHRC spokeswoman said: “We have received complaints regarding the Conservative party and are considering them in line with our usual processes. As part of our standard process, we have written to the Conservative party to ask for information in order to help assess the complaints.”

Among the allegations, the MCB dossier said comments made by the current leadership favourite Boris Johnson about women wearing burqas looking like “letterboxes” or “bank robbers” were an example of Tory MPs creating an “atmosphere of hostility towards the Muslim members of the party”.

The former foreign secretary was cleared of breaking the party’s code of conduct with the comments – a decision the MCB criticised at the time.

The group also criticised the unsuccessful London mayoral campaign of Zac Goldsmith in 2016 – when he ran against Sadiq Khan – arguing it demonstrated a “tolerance for Islamophobia” within the Conservative party.

The MCB’s letter also names Bob Blackman, the Harrow East MP, citing his membership of Islamophobic Facebook groups, his retweeting of a message by the anti-Islam activist Tommy Robinson, and his decision to host an event in parliament for the hardline Hindu nationalist Tapan Ghosh.

Blackman has said he shared the Robinson tweet in error, was added to the Islamophobic Facebook group without his knowledge and had not been aware of Ghosh’s anti-Islam views at the time.

The Conservative party chairman, Brandon Lewis, insisted the party’s approach to Islamophobia was “transparent” but did not respond when asked how many complaints it had received.