It's no coincidence that America, the most religious of the Western nations, is the land most defensive of free speech. I'm sure you wouldn't want to turn the internet over to the secular United Nations.
Think about it. Does anyone ever talk about moving to secularized Canada or Sweden or France because they have a greater tolerance of free speech there? No, they talk about moving there because the welfare checks (paid for by confiscatory taxes on others) are fatter, or because those countries don't bother much with fighting Islam, or because they have cradle to grave socialized medicine, or because they're a "gun free" environment (meaning law abiding citizens have a harder time getting guns), or because they have tougher "hate speech" laws.
That may seem contradictory but it is reality.
Will it survive? There's good reason to fear the stupidity of politically correct secularists but I think it will. We are so strong and vibrant and so capable of absorbing good things and good people from other cultures that it is hard to see how we could fail.