He wrote: '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.'
She responded: 'Alex, I find your message offensive. I am on Linked-in [sic] for business purposes, not to be approached about my physical appearance or to be objectified by sexist men.'
He emailed her an apology on Wednesday, but Miss Proudman said there was 'no acknowledgement that the message he sent was inappropriate or is sexist'.
Within hours, thousands of people were replying to her tweets, praising and ridiculing her for shaming Mr Carter-Silk.
However, it emerged yesterday that the award-winning human rights barrister has commented on pictures of men on Facebook herself to praise their looks.
She also told female friends they looked 'sexy' and 'stunning' - the same word used by Mr Carter-Silk.
On the profile of a postgraduate student at Cambridge, where Miss Proudman is on sabbatical from her chambers to study for a PhD, she wrote: 'Hot stuff!', while under an image of a long-haired male friend, she wrote: 'oooo lalala!'
Beneath photos from women, she was also happy to compliment their looks, saying to different friends: 'Oh ladies, wowwweeeeee!!!!!!', 'wow! stunning!!!!' and 'Sexy lady!'
Miss Proudman said Mr Carter-Silk's comments were particularly unacceptable because he made them in a workplace environment in his capacity as a senior solicitor.
She told the Mail: 'I have received messages from LinkedIn before but never one from a senior partner at a solicitors' firm, which is why I called it out. As a solicitor he has a duty to uphold the law, which includes sexual discrimination.'
Yesterday, she hit back after receiving a torrent of abuse online for her comments and being labelled a cause celebre for armies of 'Feminazis'.
Miss Proudman told the London Evening Standard: 'Of course I am not a man-hating Feminazi. That is an incredibly insulting thing to say and just another mechanism to silence women.'
She said the public interest in exposing sexism outweighed Mr Carter-Silk's right to privacy, and repeated her demand for a public apology, and pointed out that her boyfriend, a 30-year-old Cambridge PhD student, received job offers to work at hedge funds via LinkedIn while she got 'propositions from men'.
She added: 'There is a continuum between receiving a sexist message on LinkedIn and being discriminated against in the workplace.
'It has a huge, profound effect on women's career opportunities, making them feel uncomfortable working in male-dominated places, for example in the law. That is why I try to nip it in the bud before it escalates.'
However, in a blog on the left-wing website Left Foot Forward, Ms Proudman wrote: 'The crux of the matter is that men live and work in a brutal society, which is maintained through stratified social order based on ritual humiliation, gentleman's clubs, fights, rites of passage, sexism, and banter.
'When women enter the male realm whether law, politics, or a construction site, they find themselves in a repugnant world in which their only means of survival is by undergoing a fundamental transformation leaving them with little opportunity to make any change.'
She has been warned she has 'ruined her career' by criticising Mr Carter-Silk and speaking out about sexism. Franklin Sinclair, a partner in one of the country's largest criminal law firms, warned she could be blacklisted.
Miss Proudman, from Leek in Staffordshire, specialises in the law surrounding female genital mutilation. She said LinkedIn was seen by some members as being like the dating app Tinder, adding: 'Professionals are using LinkedIn as if it were Tinder. There are websites designed for professionals to date. There is no need to use LinkedIn.'
She did not respond last night to requests to comment on the remarks she had made about friends on Facebook.
- Daily Mail