Obviously, anyone inclined to talk about God in a public policy discussion is going to claim He is in favor of their proposal. No one is going to say, "I think God is squarely against my view, but it's worth considering nonetheless." So the fact that people invoke God's support of their policy positions is a function of human psychology, not an indicator of how God would vote were he inclined to participate in the democratic process.
That probably sounds a little flippant. I think that's because we all sense that God does not participate in the democratic process, and I think it is slightly disrespectful to the deity to suggest He would. The best we can hope for in terms of providential intervention is a bit of inspiration here or there, and perhaps a small miracle on rare occasions. I think the Second Coming is a good way off. In democratic politics, we're on our own.
Somewhat independent of that question is the issue of whether churches, or the Church, should actively comment or campaign in favor of selected public policy issues. One can go either way. On the one hand, I reject the idea that churches or religious entities should be frozen out of public debate. It would be unfair, for example, for proponents of gay marriage to make their arguments in the media and rally support for an initiative, then turn around and argue that churches (which are the foundation of conservative opposition, it seems) cannot participate in the process or offer counterarguments or campaign against it. Religion, as part of society, deserves a voice on matters of concern just like every other institution in society has.
On the other hand, Americans have always distrusted a too-active public role for institutional religion. Constitutionally, ministers aren't barred from public office, but practically speaking, people are generally uncomfortable mixing religion and politics in the same person. The more active and visible churches or religion become in political campaigns or issues, the more that latent discomfort with mingling religion and politics emerges. So churches are better off avoiding high-profile political activity. They are best advised, I think, to work indirectly or even avoid any involvement, trusting instead to the voluntary efforts of individual members to accomplish political tasks, perhaps with some gentle encouragement.
So that's my position: Churches should have a right to be active in public policy issues if they so choose, but doing so is always risky. One of the most regrettable side effects of a more active political role is the possibility of alienating members who hold different political views. Or worse, a cowboy Bishop who takes it upon himself to enforce what he sees as a mandate for political conformity by Church members. That's the best argument for keeping the Church out of politics. They've been pretty good about that in the past, but with "moral issues" becoming the focus of political campaigns lately, that is becoming more difficult.
I think it would be a real mess if the Church started making public support of anti-gay marriage initiatives, or participation in Church-supported activities designed to further that goal, a condition of good standing in the Church.