Rory McIlroy hopes Muirfield will " see some sense" and reconsider its membership policy, while the chief executive of the Ladies European Tour has described the ongoing debate about gender equality in golf as "embarrassing".
After a vote on accepting women members narrowly failed to achieve the two-thirds majority required, the R&A announced that the Scottish course would no longer be considered to host the game's oldest major championship.
McIlroy missed the cut the last time Muirfield staged the Open in 2013 and described his own play as ''brain dead'' following an opening 79, but lifted the Claret Jug at Hoylake 12 months later.
" I probably had the worst professional week of my career at Muirfield, so I don't feel that bad not going back to be honest. My memories of Muirfield aren't very good," said McIlroy, who also wants to see Portmarnock change its male-only membership policy so it can stage the Irish Open.
"They can do what they want but in this day and age it's not right to host the world's biggest tournament at a place that does not allow women to become members. Hopefully they can see some sense and we can get it back there one day.
"The R&A did the right thing. It's 2016 and we have to move with the times. It's taken long enough.
"Bigger picture, it's a great golf course, but there's so many other great golf courses that we play on the Open rota that we're not going to miss one."
LET chief Ivan Khodabakhsh cannot believe the issue is still having to be discussed.
"We are in the 21st century. Even having this discussion is embarrassing," he told BBC Radio 5 Live.
"Let's just put it behind us and move on. Certainly the Ladies European Tour won't ever have a tournament in a club which is male-only.
"Let us just put the decision of Muirfield aside as a private club. They can decide what they want to do but they can't have the exposure in the world the way they want it.
"But I have to say at the same time I'm very pleased about the R&A's decision. The R&A has been a huge supporter of the Ladies European Tour and the women's golf game.
"Scotland as well, which is hosting the Solheim Cup in 2019, has shown very clearly how they support the women's agenda and the women's sport agenda."
Muirfield has staged the Open 16 times and produced a who's who of champions, including Gary Player, Jack Nicklaus, Lee Trevino, Tom Watson, Nick Faldo (twice), Ernie Els and Phil Mickelson.
Player echoed McIlroy's sentiments, writing on Twitter: "As much as I love and respect Muirfield as a club where I won the Open, I totally agree with the R&A that staging the championship at any venue that does not admit women is simply unacceptable.
"I hope Muirfield will reconsider their position soon and continue hosting one of the world's greatest golf events."
Of the 648 members who were eligible to vote, 616 took part with 397 voting in favour of accepting women and 219 against. A total of 411 yes votes was required.
Royal Troon is the only remaining Open venue that retains a male-only membership and will host this year's tournament in July. The club is currently reviewing its membership policy with a decision expected towards the end of the year.
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."
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