Issues like gay marriage and abortion are always sensitive because they are in many ways a morality question, which is inherently subjective. If everyone has a different set of values and belief systems, how can we trust the decision of one group over another? Sure, there are things that we outlaw because most of us will agree on - murder, for example - but there are a lot of gray area out there. When it comes to abortion, for example, I may decide that it's wrong based on my values and morals. At the same time, that's a personal choice, and I don't think it should be binding against everyone.
Whenever one of these issues comes up, though, one of the standard responses is an appeal to the founding fathers' religious beliefs. "This nation was founded on Christianity", they tell me.
A good write-up in The Week a few weeks back discusses the faith of the framers in depth.
The faith that many of the Founders embraced was deism. Less a religion than a way of perceiving divinity in the world, deism is rooted in the 17th- and 18th-century scientific and philosophical revolutions of the Enlightenment. For deists, God is not a father figure who dwells in heaven and performs miracles. Rather, he is an undefined and unknowable “prime mover” who reveals himself in immutable laws that can rationally explain both cosmic and human affairs. Everything from the physical forces that govern the universe to the essential freedom of man, deists believe, are outward signs of God’s presence among us. In keeping with their deist beliefs, the Founders often refrained from calling the source of their inspiration “God.” He—or, rather, it—was “Divine Providence” and “The Universal Sovereign,” among other euphemisms.
We need to remember that religion is a man-made construct. It's an attempt to understand something so abstract that we have no chance of understanding otherwise.