It is indisputably a big political milestone. By winning the Texas Republican primary this week, Mitt Romney has finally clawed his way to his party's nomination to challenge Barack Obama. Never before has a member of the Church of Jesus Christ of Latter-day Saints been as close to the US presidency as Romney is today. And, make no mistake about it, Romney could win in November.

Romney's Mormonism is still the one thing that most people know about him, especially outside America. But does his religion matter significantly either to his chances as a candidate, as a potential president and as leader of our most powerful ally? The standard worldly wise answer to this question is to answer no. But I'm not so sure.

Those who give the standard answer make three main points. First, they say that America is a more diverse and open society now than in the past, as the election of Obama himself underlined in 2008, so the prospect of a Mormon president is less threatening to voters than it might once have been. Second, they say that Romney dealt very effectively with the issue in a December 2007 speech in Texas and that the subject has not got back off the ground to threaten him this time, even in the Republican primaries. Finally, they say that the 2012 election is about so many other issues that are far more important to voters, above all the economy, and that Mormonism is low on the list of Americans' concerns about Romney.

There is, it should be added, a fourth point, more rarely made in public. This is that, however bizarre it may be to have a Mormon presidential candidate, this is not an issue that Democrats have an interest in highlighting. That's because so many Americans – one in five – still believe that Obama is a Muslim and born outside the US. There are few political rewards from allowing the election to be framed as one weird religious guy against another weird religious guy.

All of this may be true. But we should not overlook the spectacularly weird worldview of Mitt Romney's religion. Just as the one thing that most people know about Romney is that he is a Mormon, so the one thing that most people think they know about Mormons is that they practise polygamy. But this is unfair. Polygamy certainly still exists in some Mormon sects, but the main church repudiated it a century ago – as they much later abjured the church's earlier doctrine that black people are inferior beings. It is not the movement's past but its present that matters most.

For polygamy and racism were merely nasty subsets of Mormonism's animating piece of dangerous nonsense. Underlying the whole of Mormonism, past and present, is the claim – part of what makes it a heresy to traditional forms of Christianity – that America is divinely blessed and the US divinely inspired. These claims run through the Book of Mormon, supposedly translated – with the aid of special stone glasses – by the movement's semi-literate founder, Joseph Smith, from golden tablets dug up by him in New York state in the 1820s at the direction of the angel Moroni, who then conveniently removed the plates to heaven.

Among Mormonism's core beliefs is the assertion that Jesus Christ actually visited America two millennia ago. Mormons also believe that the Garden of Eden was in America, that native Americans are part of the lost tribes of Israel, and that America is the land of Zion where the new Jerusalem will be built before the second coming of the messiah. They hold that the Declaration of Independence and the US constitution were divinely inspired documents towards that end – though it is not so clear, as the historian Garry Wills has pointed out, whether Mormons regard the constitution's later amendments, including the abolition of slavery, as divinely inspired too. George Washington and America's founding fathers have all been posthumously incorporated into the Mormon narrative and rebaptised as Mormons, as has Christopher Columbus.

There are two reasons why this view of the world matters outside the confines of the Mormon temples. The first is that it is a good old-fashioned spectacular Christian heresy. This has electoral consequences. Mormonism makes it difficult not just for agnostics but for members of other religions, including the more doctrinaire American Christians of course, to vote for someone who professes it, as Romney does. The second matters more widely. If America has divine sanction, as Mormons believe, it follows that American foreign policy has divine sanction too, and that it is answerable to its own (ie God's) rules, not those of other, by definition lesser and less blessed, nations.

Perhaps, as many believe, Romney's Mormonism will not loom large in November. There are, after all, at most 6 million Mormons. But this seems at odds with the widely accepted view that American politics, especially on the right, continue to be deeply influenced by religious and cultural divides. Romney may not flaunt his religion but his 2007 speech was an affirmation, never since denied, of the centrality of religion in American public life. And individuals matter in politics. If he becomes president, a man who believes that the existence of the United States is proof of God's higher purpose for its rulers will again occupy the Oval Office. Does Romney's Mormonism matter? You bet it does.

• Follow Comment is free on Twitter @commentisfree