The former prime minister said it was indefensible that the club, which until 2004 set the rules of the game worldwide, was still men-only after Augusta National, the most powerful club in the US, abandoned its ban on female members this week.
"If the golf club in Augusta can admit women, then shouldn't St Andrews? If they can do it in South Carolina, can we not do it in Scotland?" Brown asked.
He made the comments at the Scottish parliament festival of politics in Edinburgh, where he spoke about the key role played by Scottish political and trades union leaders in championing social justice across the UK. "I think we have to think hard and long about issues of discrimination in our own country," he said.
In 2009 the R&A was criticised by Alex Salmond, the first minister, after it ignored its tradition of inviting the principals of St Andrew's University to become members when Louise Richardson took up the post.
Soon after Augusta's decision, the Scottish sports minister, Shona Robison, urged the R&A and Muirfield, another all-male club, which is hosting the Open next year, to follow their lead. "The first minister is on record in saying that all clubs should be open to women members, but there is nothing we can do to force them. However, the direction of travel now is clearly to admit both men and women members and hopefully we will see an end to not allowing women to become members," she said.
The R&A would not comment directly on Brown's comments. It said in a statement: "The rules of the Royal and Ancient golf club at St Andrews specify a male membership and this policy remains a matter for our members determine."
In his speech to the Scottish parliament, Brown also urged Scottish voters to reject independence and stay within the UK to protect common standards on social justice, welfare rights and the minimum wage.
He said many Scots had played a significant role in creating the British welfare system and the labour movement. The National Union of Mineworkers was partly founded by a Scot, Alexander Macdonald; Keir Hardie helped found the British Labour party; and Tom Johnston, Scottish secretary in the UK cabinet during the second world war, laid down the foundations for the NHS.
Taking inspiration from Scottish ideas, they had pressed for shared UK-wide standards on factory safety, minimum wages, national insurance and free healthcare because they believed in equal rights and equal standards of social justice across all parts of the UK, Brown said.
"Were they wrong? Were they betraying Scottish values and traditions or were they not realising Scottish values and traditions? They did it because they believed that the principles of social justice apply to everyone, they did it because they believed it was right to pool and share resources, and they did it because they didn't want to see conditions in Scotland undermined because there was a race to the bottom because different parts of the UK were offering different standards and conditions."
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