A lot of misinformation in this thread.
1. White Evangelical Christians make up less of a percentage as a whole of the nation, but Evangelical Christianity is still out there. Studies show that Catholic and mainline protestant congregations have experienced population loss (particularly Episcopalians), but evangelical/non-denominational churches are still filling their pews. For every white millennial who drops their religious observance, there's a white ex-Catholic, Hispanic, or some other immigrant-type to take their seat.
2. Most conservative Christians/evangelicals wanted Ted Cruz, not Donald Trump. Well, a good chunk also voted for Trump, and a few voted for Carson and Rubio, and that was enough vote-splitting in an already wide field for the fairly secular New Yorker to bust through the social conservative firewall. The Republican Party is more diverse than some people think it is.
3. America is a different animal from Europe. Europeans stopped attending church when large secular governments took over their war torn, cynical countries and they still had their nationality, ethnicity, deep history, and soccer hooliganism to keep themselves united and happy. While it's true that many Americans have ditched church, many more (white) Americans continue to keep religion as a central part of their identity, social life, and worldview. This is because America is bigger and more diverse than Denmark, and people need some sort of community/identity/faith/something-to-keep-them-happy. For this reason, I don't see religion ever completely tanking in the USA the way it has in Western Europe, even amongst white Americans.
4. It is also important to note that Christians/churches are having a crisis from within. The future of Christian doctrine itself hangs in the balance. You have the obvious moderation in tonality and tactics from Falwell/Robertson/Graham to the Rick Warrens of today. Then you have the Joel Osteens, the accusations of the watering-down or outright liberalization of Christianity. Christians have their own "cuckservatives" to fight on the inside (read Dalrock's blog). I've met some unabashed progressive Christians who would not disagree on one major issue with the average secular American leftist. Then you have the ones who still want to fight the culture wars. And then there's the ones who are personally traditional/conservative, but just plain apathetic on politics. Whatever happens inside of churches and what a "Christian" looks like in the future, I think the Religious Right is definitely less of a united political force today, so OP's proposal of an avowedly secular Republican Party sounds plausible on paper, but even Donald Trump had to say "we're saying Merry Christmas again folks!".
I see the future leaders of American Christianity being people like Rachel Held Evans and Matthew Vines as opposed to Tony Perkins, Jerry Falwell Jr., and John Piper.
