"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 in the pro-am before the Scottish Open 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.
Dawson said the R&A 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 gender-exclusive clubs in Britain, 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."