The key immigrant groups that the Republican Party needs to win over don't believe in the "our religious beliefs are under attack" meme though.
Quite simply put though, they are under attack. Therefore I can only see improvement in this area should the GOP commit some resources to educating the public about this. If the GOP championed liberties for all instead of created wedges it would be a lot easier to woo these immigrants who quite frankly remain in the dark about the long and contentious public debate on abortion, gay marriage, and so on.
But in Canada, ironically enough, things have been steadily swinging to the right on social issues since the 1980s or so. I honestly wouldn't be surprised if bringing back the death penalty and banning abortion became viable political moves in ten or twenty years.
The difference, though, is that there is a sizable portion of the GOP electorate that basically votes on the religious issues that they need to hold on (especially in the south and midwest), whereas in Canada such voters are basically marginal and only exist in safe Reform/Conservative areas like Alberta and the interior of BC.
Opposition to UHC in the US has a pretty long history, though. IIRC Truman tried to get something along those lines passed in the 1950s, but at there were various factors that made it very unpopular; people had already overwhelmingly voted against the government control of the war years (ending rationing and so on), and at the time American healthcare was quite affordable and of very high quality to boot.
News of the GOP's demise, however, is greatly exaggerated. Yes, immigration means that the "old white male" demographic they have locked up won't be enough to help them in the future, but (A) Hispanics aren't really out of their reach and (B) the Rust Belt has been swinging in their direction enough to mean they can practically just expand their share of the lower-middle class white vote to make up for future problems. In the long term their views will change by political necessity and I suspect will be radically different by 2020, but they are hardly endangered right now.
Anyway, the GOP wouldn't really benefit from dropping the social issues altogether since such a huge portion of their electorate relies on those. However, they would stand to benefit by not making otherwise irrelevant issues like gay marriage hills to die on since they'll probably lose fights like that in the long run. Abortion is something they might have a bit of a better chance on, seeing as how public opinion has been going in the pro-life direction for a while.
Honestly, I see the "successful" GOP of 2020 being somewhat socially conservative if less so than presently with a libertarian bent. Dropping the drug war would probably let them appeal to the urban poor and Hispanics a bit better, and I'd say their present pro-war/interventionism stance isn't going to last either.
