Three-time major champion Padraig Harrington has backed the decision to remove Muirfield from the Open rota following the club's vote to continue refusing women members.
A two-thirds majority was required to alter the Scottish club's constitution but only 64 per cent voted in favour and 36 per cent against on Thursday.
Muirfield has been told by the R&A it will no longer host the Open Championship and Harrington, who won the tournament in 2007 and 2008, believes the correct decision has been made.
"The R&A have done the right thing for the bigger picture," Harrington said.
"They lead golf on a world stage and they have a responsibility to general society not to condone it or to drift along as if it's not happening.
"Muirfield may go ahead and say we want just be a small golf club and mind their own business and that's fine, but minding their own business isn't holding the Open is it?"
A letter leaked to the Scotsman this week suggested Muirfield members were concerned about women players feeling "uncomfortable", as well as their speed of play.
"Generally ladies play a lot faster than men," Harrington said. "Every lady I've ever seen on the golf course seems to want to rush around, they are so anxious about that end of things.
"There is no reason why it can't work and in this day and age, who would want to be in a golf club with only men?"
Scotland's Paul Lawrie, who won the Open at Carnoustie in 1999, believes Muirfield's view are outdated.
"The R&A are right to do what they've done," Lawrie said.
"How can you not let ladies be a member of a golf club? It just doesn't make any sense. Imagine not being able to take your wife into the golf club. It's just not right."
Six-time major champion Nick Faldo added: "While I accept that private clubs have the right to create their own policies, I fully support the R &A and Martin Slumbers' many forward thinking initiatives toward inclusiveness and growing the game of golf around the world."
Catriona Matthew, who won the Women's Open in 2009 at Royal Lytham and lives only a few miles from Muirfield in North Berwick, tweeted: "Embarrassed to be a Scottish women golfer from East Lothian after that decision."
Thomas Bjorn, chairman of the European Tour's tournament players' committee, was left in disbelief by the vote.
"It's a shame that one of the best golf courses in the world is off the Open rota. I'm gobsmacked," Bjorn said.
"From a golfing perspective it's a shame, especially for the bigger picture. People can have their little clubs but if you want to be in the public eye then I'm sorry, I think we're past that."
The announcement was also greeted with anger by those outside the game, with Scotland's First Minister Nicola Sturgeon one of the first to express her disapproval.
Responding to the news on Twitter, Sturgeon said: "Scotland has women leaders in every walk of life. It is 2016. This is simply indefensible."
Minister for Sport David Evennett said: "It is an extremely disappointing decision from the members of Muirfield. We want to encourage more women to get involved in sport and this sends out completely the wrong message."
Royal Troon is the only remaining Open venue that retains a male-only membership and is due to host this year's tournament in July. The club is currently reviewing its membership policy.
Local MP for East Lothian George Kerevan said: "This utterly selfish move sends entirely the wrong message to the rest of the world about our county, as well as undermining years of collective effort and public support to attract visitors to Scotland's historic "golf coast" in East Lothian."
Ruth Holdaway, the chief executive of the Women In Sport charity, described the decision as "ridiculous" and called for current members to protest by ending their affiliation with the club.
Holdaway said: "In this day and age it's completely ridiculous and it's also commercially a really bad decision."
She had earlier said: ''Women in Sport suggests that Muirfield members who feel this vote result is wrong now consider terminating their membership, boycott the club and go and play their golf elsewhere in an environment that welcomes us all.''
SNP Equality spokesperson Angela Crawley MP added: "Muirfield has caused great embarrassment for Scottish Golf, and Scottish sport more widely when great efforts are being made across Scotland for sport to be made more inclusive."
The Royal and Ancient Golf Club at St Andrews chose in 2014 to admit female members for the first time in 260 years and Royal St George's in Kent also changed its male-only policy last year.
Muirfield last staged the Open in 2013, when the competition was won by American Phil Mickelson.
Henry Fairweather, captain of the Honourable Company of Edinburgh Golfers (HCEG), which owns and runs Muirfield, said policy is decided by the club's members.
"The Honourable Company is a members' club, and, as such, the members decide the rules of the club, including its membership policy," Fairweather said. "Women will continue to be welcome at Muirfield on the course and in the clubhouse as guests and visitors, as they have been for many years."
R&A chief executive Martin Slumbers said in a statement: "We have consistently said that it is a matter for the Honourable Company to conduct a review of its membership policy and that we would await their decision.
"The R&A has considered today's decision with respect to The Open Championship. The Open is one of the world's great sporting events and going forward we will not stage the Championship at a venue that does not admit women as members.
"Given the schedule for staging The Open, it would be some years before Muirfield would have been considered to host the Championship again. If the policy at the club should change we would reconsider Muirfield as a venue for The Open in future."