I think the Republicans have come to a fork in the road where they have to make a choice: be the party of small government, or embrace the Christian Right? I think conservatism will never die out, because it's partially a personality type in humanity (as is liberalism), but parties and party identities can die out. Right now, the Republicans are diluting their small government message with a lot of social conservatism that, more and more, consists of issues that are political losers. In the past, being for legislating away rights to gay marriage and abortion, for example, could win you more votes. Polls show that the majority of the American people are either now against such legislating away of right or will be soon. It's also contradictory to the message of small government...it can correctly be characterized as "Small government except for the things we don't like."
I think a conservative party that ditches the deep social conservatism and sticks to fiscal conservatism will do strongly. When people say the country is "center-right," it's now important to qualify that: it's center-right on fiscal issues but more and more center-left on social issues. Democrats are probably more left of the population on fiscal issues, but Republicans are currently too far to the right of the population on social issues and trending away from the population. If they continue that path, they'll marginalize themselves. It'll still play in deep south states, but it won't win national elections.
I don't know if the Republican party will realize this in time for 2016. The GOP primary season took a guy who governed as moderate Republican in Massachusetts and forced him to run to right of every lunatic (read: Rick Santorum, Rick Perry) in the primary in order to win. If the GOP keeps doing that, they'll keep losing national elections. At some point, either the Republican party will change to focus on fiscal issues while ditching the social conservatism baggage or else a new conservative party will arise who does.
Personally, I find the current Republican party repugnant, but I don't find fiscal conservativism repugnant. While I'm obviously much to the left of the center, I think the Republicans have a similar problem, to a lesser extent, with a lot of people in the nation.