Later, when someone asked McIlroy if the players had been advised not to comment, he was more forthcoming.
"I just think it's something that a lot of guys don't want to get themselves into because it's quite a controversial issue," he said. "It's something that shouldn't happen these days. It's something that we shouldn't even be talking about."
Ernie Els said it's "weird" that some clubs won't admit both sexes, while Luke Donald said "we'd love to see these policies be a bit more inclusive."
"Wherever the governing bodies decide to play a tournament, it's my job to turn up and be ready and play, and that's what I am going to," Donald said. "I think the R&A is certainly trending in the right direction."
But the leading Scottish politician won't be attending this year's event in protest.
"I just think it's indefensible in the 21st century not to have a golf club that's open to all," said Scottish First Minister Alex Salmond, a huge golf fan who played a round with Phil Mickelson last week.
Two British government members — Maria Miller, the secretary of culture, media and sport, and sports minister Hugh Robertson — have also turned down invitations to attend, though Robertson downplayed the impact of his decision.
"I haven't had a call from the R&A saying we are going to change our policy next year, nor would I really expect one," the minister said.
Indeed, Dawson said the organization would not give in to political pressure.
"We've been through over 250 years of existence without getting into political comment, and I don't really intend to break that rule here," he said. "We've got politicians posturing; we've got interest groups attacking the R&A, attacking the Open, and attacking Muirfield."
While conceding that some changes are likely, Dawson made clear he believes the issue has largely been manufactured by those who don't necessarily have the best interests of the game at heart. He claimed there are very few clubs in Britain that allow only one sex, and that half of those are female-only.
"You can dress it up to be a lot more, if you want," Dawson said. "But on the Saturday morning when the guy gets up or the lady gets up and out of the marital bed, if you like, and goes off and plays golf with his chums and comes back in the afternoon, that's not on any kind of par with racial discrimination or anti-Semitism or any of these things.
"It's just what people kind of do."