The House of Commons speaker said he “respectfully” did not accept claims made by energy minister Claire Perry.
She wrote to him following an appearance at the despatch box on Tuesday.
During MPs’ questions to the business department, Mr Bercow told Ms Perry to “please resume your seat”, adding: “I’m afraid dilation is not in order today.”
Ms Perry attempted to laugh off the comment, opening her next answer by quipping that the last time someone had accused her of dilation was when she was in labour.
But, Mr Bercow responded swiftly: “What is required is a brief answer and brief question – no dilation”.
In the letter, obtained by The Sun, Ms Perry notes that she had written previously to Mr Bercow about his “aggressive and disrespectful tone” towards female colleagues.
She also claimed that “more than a dozen” fellow MPs had told her they were surprised by the speaker’s “demeaning” remarks towards her.
Ms Perry wrote: “During BEIS [Department for Business, Energy and Industrial Strategy] oral questions today, you used a tone and language to admonish me for giving over-long answers that seemed both overly-aggressive and particularly sexist.
“I fear a pattern is emerging where you are finding it increasingly hard to interact respectfully with female MPs on the government benches.
“At a time when we are dealing with an endemic problem of bullying and violence – both physical and verbal – against women in the UK and elsewhere, it is not acceptable to have a presiding officer of our precious Home of Commons who appears to have a ‘woman problem’.”
Mr Bercow responded thanking Ms Perry for the letter. He said: “While – respectfully – I do not accept the points made, perhaps it would be nice to meet for a ‘clear the air’ cup of tea as soon as possible?”
A spokeswoman for the speaker’s office claimed Mr Bercow had used the term dilation “many times before”.
According to Hansard, the Buckingham MP has indeed used the term “dilation” more than a dozen times in recent years.
It is the latest in a series of criticisms levelled at Mr Bercow in recent months.
As well as claims of bullying staff, the speaker was also accused of calling Leader of the House Andrea Leadsom a “stupid woman”, in off-microphone comments in the Commons.
Mr Bercow admitted using the word “stupid” in a “muttered aside” but insisted it was not directed as Ms Leadsom.