The school, which is the alma mater for high-profile political and royal figures including 20 prime ministers and princes William and Harry, is embroiled in a row over freedom of speech after a teacher was dismissed for a lesson he posted on YouTube.
Mr Williamson refused to offer a comment on the situation, telling BBC Radio 4’s Today programme that it is “an ongoing disciplinary procedure” he did not think he should get involved in.
He added: “I’m a great believer in freedom of speech, a great believer in political impartiality in schools, but I’m very much focused on making sure we deliver the very best education for all those in state schools, where I’m firmly responsible.”
Responding to a suggestion that admitting girls to Eton might help the case, Mr Williamson, who attended Raincliffe Comprehensive School in Scarborough, said: “I’d be very much in favour of Eton taking girls, I think it’d be a good step forward.”
In response to Mr Williamson’s comment, Jacob Rees-Mogg said Eton “works very well” without admitting girls.
The Commons leader, who himself attended Eton, told MPs: “Much though I might like my own daughter to go there, I think it works very well as it is, thank you very much.”
Boris Johnson’s official spokesperson said single-sex schools are an “important” part of the education system, but added it would support Eton if it decided to admit girls.
The spokesperson said: “We have said consistently single-sex schools are an important part of our diverse education system and it is right that parents have the opportunity to make decisions about the type of school that their children attend.
“If Eton were to become a mixed-sex school we would obviously support that decision, but it is a matter for Eton as an independent school to set their own admission criteria.”
The row at the prestigious school centres around the sacking of Will Knowland, an English teacher at the 580-year-old school who posted a video on YouTube claiming that patriarchy was rooted in biology.
Eton was accused of suppressing free speech after Mr Knowland’s dismissal, which it denied and said it was a disciplinary measure for repeatedly failing to remove the video when asked.
Lawyers told The Times the school made the right decision, as allowing Mr Knowland to continue after expressing “views that society found unacceptable” could have left Eton open to an investigation from the Equality and Human Rights Commission.
However, things escalated after Dr Luke Martin, who taught divinity at Eton, resigned in protest of the dismissal and accused the school of trying to “indoctrinate” its students with a “so-called ‘progressive’ ideology”.
In a letter to Eton’s vice-provost, first reported by The Telegraph, Dr Martin said the ideology, which “claims to be inclusive, tolerant, and kind” was similar to forms of religious fundamentalism that meant “if you disagree with it, you’re excluded; if you think differently, you’re not tolerated; and if you raise objections, you’re mocked or face formal discipline”.
Support for Mr Knowland has been growing among pupils, who last week published a petition accusing Eton of “institutional bullying” and claimed it was a “gross abuse of the duty of the school to protect the freedoms of the individual”.
The petition garnered 2,300 signatures from previous and current pupils. A crowdfunding page launched to raise money for a potential employment tribunal Mr Knowland may face if his internal appeal against the decision to sack him is rejected drew £50,000 in donations.