"Happiness" actually had a different definition, way back at the time of our founders. Like many words in our lexicon, they evolve and change over time. "Happiness" was one of them. Go back and look it up. You'll see one of the principle definitions of happiness is "to do the morally right thing." God gave us rights to life and to freedom to pursue His will. That's what the moral foundation of our country is.
So Santorum is basically saying that everyone has to believe in his god. An atheist like me can't pursue happiness because I don't believe in the moral code that he does. That's an extremely perverted view of the very foundation of this country.
Conor also makes a good point in pointing out that the famous phrase was almost "property" instead of "pursuit of happiness". That's because Jefferson took the line "life, liberty, and property" from John Locke. Jefferson and the founders didn't get these ideas from god. They got them from men like Locke and other political philosophers. Those men are where rights come from, not god.
Another point I want to make is that while the DOI is nice, it's not the law. The Constitution established our form of gov't and the law of the land. The DOI is us telling England we are our own country. And then it's a lot of great rhetoric. And that's why Santorum and other politicians like to use it. It gives them a chance to pick and choose the rhetoric that conforms to their beliefs. But what really matters as to policy is the Constitution. And Santorum isn't fond of quoting that because it says he can't impose his religion on the nation.