1. I can't vote for Romney because he's a Mormon
Some have actually reasoned a saner version of this argument: they say electing Romney to President assists Mormonism in becoming a mainstream religious option. The less-cogent versions of this range from claiming to shun idolaters to disbelief that a non-Christian can actually make reasonable judgments about justice and law.
Well, first of all, this sort of reasoning ignores the meat of Romans 13 almost as if it was never written. There Paul says this:
Let every person be subject to the governing authorities. For there is no authority except from God, and those that exist have been instituted by God. Therefore whoever resists the authorities resists what God has appointed, and those who resist will incur judgment. For rulers are not a terror to good conduct, but to bad. Would you have no fear of the one who is in authority? Then do what is good, and you will receive his approval, for he is God's servant for your good. But if you do wrong, be afraid, for he does not bear the sword in vain. For he is the servant of God, an avenger who carries out God's wrath on the wrongdoer. Therefore one must be in subjection, not only to avoid God's wrath but also for the sake of conscience. For because of this you also pay taxes, for the authorities are ministers of God, attending to this very thing. Pay to all what is owed to them: taxes to whom taxes are owed, revenue to whom revenue is owed, respect to whom respect is owed, honor to whom honor is owed.
But it actually gets worse for those who are worrying about legitimizing any other theology or religion or way of life: in Paul's view, God has instituted the governments we find ourselves subject to, and he says they serve God. Let's stipulate right away that they may serve God poorly, but in this case this is not Paul's point at all. Paul's point is that the institution of government is actually God's ordinary means of looking out for justice and judgment -- and that one doesn't need to be a believer to make one into a decent magistrate.
Paul says that explicitly about the Roman government -- which, let's face it, is barbaric by our standards. The kind of morality the average Roman would ascribe to would be absolutely wanton by our post-Christian standards. Yet somehow the Mormon view of morality is not going to work for an American magistrate? You know: it was the Mormons who were the major backers for the California initiative known as Prop 8 a few years ago. And the official teaching of Mormonism on the 10 commandments is easily summed up: "Obedience to these commandments paves the way for obedience to other gospel principles." That's pretty lousy Christian theology, but for a civil authority I think we would probably rather have someone who is a matter-of-fact works-righteousness guy than someone who thinks it ain't what you do but the way that you do it.
If a Roman could be someone about whom Paul could say what he said in Rom 13, don't you think a Mormon would be a more-likely minister to do what is right in the face of justice?
That deals with the question of whether or not an unbeliever is capable of being a sound ruler. What about the question of assisting a cult in becoming more mainstream? Listen: that sounds very high-minded and God-glorifying -- until we start to think about all the things we have to give up which, frankly, make things that are non-Christian into socially-acceptable practices. We'd have to give up the internet, for starters; we'd have to give up our iPhones. We'd have to give up books. We'd have to give up Capitalism and Democracy.
Let's be as clear as possible here: maybe it wouldn't really be a loss to give up all the things which are not explicitly Christian for the sake of making sure we are not accidentally endorsing things that are false gospels. Maybe that is what we are actually called to do as Christian. That's what the Anabaptists believe, and that's not generally seen as a strike against them until we start talking about medicine or electricity. Maybe that is what we're actually called to live like.
That's two or three good theological reasons that this objection doesn't work out. There is one "America" reason this doesn't work out, a reason from political philosophy. Most of you reading this are baptists, and as baptists of some sort, you gladly, gratefully embrace the idea of freedom of religion. You may or may not remember the history and results of the Half-Way Covenant among the Puritans, but you know that one of the reasons it was a flop is because it confused the necessity of the church to be filled with believers with the necessity of the civil government to act justly toward men. The foundations of it didn't understand Rom 13 at all -- and it made church membership the necessary condition of civil rights (particularly voting).
Because that attempt to maintain the unity of church and state failed (as the prime example, but not the only one), our political heritage inherited the right to freedom of religion -- that a man can practice his free expression of religion without the Government telling him what he must or must not believe, and that the Church cannot dictate whether or not a man is rightly seen as a citizen.
Voting for any man does not affirm that you accept his religious expression, or his systematic theology: it affirms that you accept his right as a citizen to run for office. If you forget that, you might need a refresher course on basic American civics.