I think it would be a mistake to assume a reconstituted opposition party (whether Republican or other) would be uninfluenced by the Religious Right.
For better or worse (and I think worse), the Religious Right is a significant segment of the voting population. By significant, I don't mean majority. I don't even mean plurarilty. I think it's probably somewhere around 20%.
That's a sizeable chunk of votes. Most importantly, what motivates them -- abortion, gay marriage, immigration, school vouchers, etc. -- is something most voters find to be a tertiary (at best) consideration. The primary consideration for most people is the economy, and secondarily military prowess. When we're attacked or at war, the second consideration becomes co-equal or, briefly, more important, than the economy.
The problem recently is that the Religious Right has had its expectations too high. They wanted their influence to be primary, when really, based on their numbers, they really should be content with influencing judicial appointments and filibustering movement in the federal government on the issue of gay marriage and immigration.
But they didn't. They decided they were kingmakers and the kingmakers wanted someone who would put their issues first. They disliked McCain, not because he disagreed with them substantively (he disagreed substantively only on immigration), but because he seemed to think of those conservative issues as less important as things like healthcare, foreign policy, and campaign finance. So they supported him tepidly and essentially abandoned him to the candidates. (The rest of the populace abandoned McCain on the issue of the economy.)
All the right-wing needs to do is go back to their role in the Reagan Coalition -- quietly accepting a slow transformation of the judiciary, and an occasional Cabinet appointment. Then they can bring back the economics voters who are willing to stomach the Religious Rights' social agenda.
They will be back. This is a democracy and the numbers dictate the politics. And the Religious Right simply has the numbers to influence any party that rises in opposition to the Democrats. After all, the Whigs fell apart on the issue of slavery. The abolitionists didn't disappear when the Whigs fell apart. They joined the nascent Republican Party.
The resurrection of the party will require two things.
First, Obama-fatigue. Right now, that seems pretty remote. Obama is popular. Chances are, the recession will be over by his next election and he can (rightly or wrongly, it doesn't matter as far as politics is concerned) claim credit for the resurgence. He seems less likely to engage in the same sort of personal scandals that plagued Clinton's tenure. But it's still early. Who knows what may come?
Second, the Republicans have to find an economic theory that counters the Democrats. It will happen. Invariably, the ruling party overreaches. How will that manifest? Union resurgence? Nonsensical regulations? Trade barriers? Overfederalization? Only time will tell, but, eventually, it will happen and the nation will begin looking for someone to take an axe to whatever the Democrats do wrong. And then all it takes is a charismatic Republican who looks like he can "set things right" again and it all starts over.
Then we can start speculating about whether the Democratic Party is going to disintegrate.... again.