A vote at the East Lothian club failed to rally a two-thirds majority of members behind allowing women to join.
The club has been barred from hosting the Open Championship as a result.
The first minister said the club had the right to set its own rules, but said the decision was "wrong" and "damaging".
Prime Minister David Cameron has also criticised Muirfield, and backed the decision to bar it from hosting the Open Championship.
Mr Cameron told LBC Radio: "My general rule is that sports clubs should be totally open to both sexes, and it's outdated not to do that, particularly if you think that you're up to hosting important championships."
A majority of members taking part in the vote at Muirfield backed allowing women to join, but did not achieve the necessary two-thirds of eligible members to push the move through.
Golf's governing body, the Royal and Ancient, said it would not stage the Open "at a venue that does not admit women as members".
Ms Sturgeon told BBC Scotland that she was "disappointed" by the "indefensible" decision.
She said: "Muirfield is a private club and they're in charge of their own rules and regulations, and I accept that, but this is 2016.
"Scotland has women leaders in every walk of life, in politics, in law, in business and everywhere else. I think this decision is wrong, and I hope there's a way of of looking at it again and overturning it.
"As well as being wrong, it's damaging to Muirfield as a club - I want to see the Open played at Muirfield, it's a fantastic golf course, so this really is a regrettable decision."
Ms Sturgeon rejected the suggestion that the decision had any wider message about Scottish society.
She said: "The fact that I'm sitting here in the first minister's office says that Scotland is a country that values the contribution of women.
"There are still barriers to women in every walk of life, but women also make progress in Scotland. Scotland is a welcoming place for women to progress in, so I don't want this decision to be seen as as reflection on modern Scotland, it's a reflection of a minority of members at Muirfield."
Other politicians have also been quick to condemn the decision.
Scottish Labour leader Kezia Dugdale said the golf club's decision was "hugely disappointing".
She added: "I would urge Muirfield Golf Club to think again and find a way to reverse this decision, otherwise it will rightly find itself ostracised from the Scottish golfing community.
"If ever a decision highlighted how much we need to keep up the fight for equality in Scotland this is it."
Scottish Conservative MSP Rachael Hamilton, who had campaigned for allowing women members ahead of the vote, said it was "very disappointing".
She said: "I understand that there are traditions involved here, and the members knew fine well what the consequences of this decision would be.
"But we have to move with the times, and this will be seen widely as a very outdated attitude for a sporting club to have."
Alison Johnstone  of the Scottish Greens, said: "There is simply no excuse for such a backward policy, and it's galling to see such a large number of this club's members voting to bring it into the 21st century yet they are blocked due to falling just short of a two-thirds majority.
"If the management of Muirfield had any sense, they'd acknowledge the strength of support from their members for a change in policy and they'd do the right thing to prevent further damage to the reputation of their venue and Scottish sport."
Nicola Sturgeon has called for the "indefensible" decision of Muirfield Golf Club to reject women as members to be overturned.