The British golfer said he was targeting "the haters" and "girly boy trolls" after becoming tired of incessant abusive messages on Twitter.
Maria Miller has raised concerns with BBC chiefs over John Inverdale's comments regarding Wimbledon winner Marion Bartoli's appearance.
Justin Rose has told ITV News that victory for him at next week's Open at Muirfield would cap a spectacular summer for British sport.
The Open resumes this morning at the Muirfield golf course in Scotland.
On his LBC 97.3 phone-in show, Mr Clegg said: "I was just dismayed and incredibly surprised to hear this still goes on in this day and age.
"I find it so out of step with everything else that's happening in the rest of society. It just seems so old fashioned and so anachronistic to just have a golf club saying that just because you are a woman you can't be a member of that club.
"I think many people will just shake their head and say 'how on earth is this still possible in this day and age?'."
He added: "I'm dismayed the club does not accept women as members. I find that inexplicable in this day and age, I really do."
A growing row on sexism has been ignited within the world of golf ahead of tomorrow's Open Championship.
Organisers have been criticised by politicians and the world number two, Rory McIlroy, for holding the competition in one a club that does not allow women members.
Sports Correspondent Natatlie Pirks reports from Muirfield.
Peter Dawson, the chief executive of golf's governing body, Royal and Ancient, has defended the decision to hold tomorrow's Open Championship at a men-only club.
Asked by ITV Sports Correspondent Natalie Pirks why racism was unacceptable within the sport but sexism was clearly acceptable, he said:
"It is just a way of life that some of these people like, and realistically that is all it is."
The world's number two golfer, Northern Ireland's Rory McIlroy, has spoken out against the club he is due to play at tomorrow for their policy of not allowing women to become members.
Speaking ahead of the Open Championship at Muirfield he said he could see why his fellow competitors have not spoken out against the "controversial issue", but that did not stop him:
"It's something that shouldn't happen these days. It's something that we shouldn't even be talking about so I guess that's one of the reasons that a lot of people don't want to talk about it.
"I just don't think it is even something is a real issue any more. Obviously it is an issue in some golf clubs. But in terms of life in general men and women are treated equal for the most part these days. And that is the way it should be."
The chief executive of Golf's governing body, Royal and Ancient, has defended the decision to hold The Open championship at Muirfield, a men-only golf club. Speaking to reporters he said:
"The media, with seemingly boundless energy and enthusiasm, are giving out the message that this is an issue and that such clubs (like Muirfield) should be condemned to extinction and we shouldn't be using one to stage the Open Championship. [...]
"We have got, as you mention, politicians posturing. We have got interest groups attacking the R and A, attacking the Open and attacking Muirfield. [...] To be honest, our natural reaction is to resist these pressures because we actually don't think they have very much substance.
"But we would like to stress that we are not so insular as to fail to recognise potential damage that campaign's like this can do to The Open championship."