I was started thinking along these lines by an article in Spectator magazine last fall, essentially discussing how "moral relativism" has become more a conservative bugaboo than a real thing. Long story short, there are people who claim to be relativists, but no one who can consistently hold to it. You know the drill: Dogma is bad, you can't categorically say that something is wrong, unless it's racism, sexism, homophobia, blah blah blah. It just goes to show how liberalism is a completely contradictory "philosophy," if you can even call it that. In any case, pure relativism is losing ground with the public and even pop culture, as a look at the popularity of principled heroes at the movies will tell you.
No, the real problem, according to Spectator, is a particular kind of morality we modernites are susceptible to: utilitarianism, for lack of a better word. What this simply means is that it is possible, still, to make objective moral statements. BUT, they can only be made if based on statistical or technical data. Want to say something is "good" or "bad" for society? Find study X or survey Y to back you up. Once you have the numbers on your side, then you're getting somewhere; otherwise, you're just fishing in the dark.
There are several reasons why this is bad for conservatives.
First, it makes argument in general a very dicey thing. As the whole healthy/unhealthy food studies prove, there is rarely such a thing as conclusive scientific proof on a social issue, of any kind. Take gay adoption, for example (something closely tied to the larger gay marriage debate); no sooner can I find a study showing that children raised by gay parents tend to do worse than those raised by straight parents than my debate opponent can put out one saying the former turn out just hunky-dory. In cases like this, desirable as such statistical information can be to buttress your argument, relying strictly on a scientific basis for moral points doesn't really do much to advance the debate. The veracity of the science itself just comes under fire.
Furthermore, this kind of approach constrains our battlefields. In arguing on whatever subject, if we restrict ourselves to statistics alone in supporting our position, we make ourselves dependent on what has and hasn't been done on it, not to mention what can and can't be done on it. Illegal immigration and drugs, for example, are subjects inherently somewhat secluded from outside analysis, so we can't argue with fully accurate data at our backs.
But there's a broader way in which this is problematic. Conservatives (and by extension the GOP) have often been branded "the stupid party," people who cling to tradition and prejudice to make decisions. There is truth to this; yet I don't consider that an insult necessarily. As a political philosophy following in the footsteps of Burke and others, conservatism is not, in the main, concerned with purported scientific "laws" of society--even valuable market-based ones such as Smith's and Hayek's--but with proposals about how we should act, individually and collectively; proposals which cannot be proven by social science. The statement that change should not happen for its own sake, for example, is one most conservatives would agree with, but you can't whip out an academic study confirming or denying this. Nor can you produce one showing that the collected wisdom of our ancestors should be considered when making a political or economic decision.
At bottom, conservatism (like other philosophies) asks questions about life, liberty, community, etc. And while the natural and social sciences can inform us about portions of those concepts, they can't tell us why those concepts are good and desirable in themselves. Nor can they tell us everything about how to best pursue and preserve those things. We have to be okay with the fact that reason alone can't answer all this. Or at least, the overvalued pure empiricism we rely on can't.
I don't have a concrete suggestion on how conservatives should proceed with this in mind. Ours is a society extremely wedded to what can be empirically proven, and nothing else. But as we go forward, we need to keep in mind that this is not the only branch of human knowledge, and maybe not even the most important one. We ought to look for a way to demonstrate that what we can't quantify has just as much to say about life and society as what we can.