"I have received messages liked that before, commenting on my appearance and using LinkedIn essentially as Tinder instead of high-lighting women's professional skills," she reportedly told the newspaper's Sophia Sleigh.
"My partner gets messages asking if he wants a job at hedge funds, I get propositions from men asking me out.
"I want a public apology."
Mr Carter-Silk's full 9.16am message said: "Charlotte, delighted to connect, I appreciate that this is probably horrendously politically incorrect but that is a stunning picture !!!
"You definitely win the prize for the best LinkedIn picture I have ever seen. Always interest to understant (sic) people's skills and how we might work together.'
Responding around nine hours later, she wrote: "Alex, I find your message offensive.
"I am on Linked-in for business purposes, not to be approached about my physical appearance or to be objectified by sexist men.
"The eroticisation of women's physical appearance is a way of exercising power over women. It silences women's professional attributes as their physical appearance becomes the subject.
"Unacceptable and misogynistic behaviour. Think twice before sending another woman (half your age) such a sexist message."
Miss Proudman, who is the same age as Mr Carter Silk's daughter , has been received a mixed reaction on social media.
Tweeting today, the barrister with Michael Mansfield QC, wrote: "Will endure misogynistic backlash that accompanies calling out sexism in hope that it encourages at least 1 woman 2 feel she doesn't need to take it."
She also posted: "Widespread, casual, demeaning behaviour directed towards women is a form of social policing, gender control & a hidden form of social violence."
In a statement released through City law message board RollOnFriday, Mr Carter-Silk, who lives in Cambridge claimed his comments had been misconstrued.
He said: "Most people post pretty unprofessional pictures on LinkedIn, my comment was aimed at the professional quality of the presentation on LinkedIn which was unfortunately misinterpreted.
"Ms Proudman is clearly highly respected and I was pleased to receive her request to linkup and very happy to instruct her on matters which [are] relevant to her expertise, that remains the position."
His employers, Brown Rudnick, later said they were aware of Mr Carter-Silk's comments.
They said: "We have apologised for the offence caused and have no further comment to make."
Meanwhile, women have come forward with their own examples of workplace sexism.
- Daily Mail