No, but conservatives do believe that people are ultimately responsible for their behavior. Why does O'Reilly and gay activists angerly insist that it is impossible for someone to stop behaving gay? It really boils their blood pressure at any suggestion that being in the gay lifestyle is anything but genetic.
If you're a conservative, the concept that special rights and privileges ought to be granted to people who engage in specific sexual behaviors should be repugnant to you.
Conservatives believe, at the very least, in equality before the law - not the creation of privileged groups.
"Anti" not in a hating of the person sense but yes conservative means both social and fiscal. Conservatism absent social moral construct is called Libertarianism.