How we cite our quotes: (Paragraph.Line)
Quote #1
But if we wish to continue to be distinctively American, we must continue to make that term comprehensive enough to embrace the legitimate desires of a civilized and enlightened people determined in all their relations to pursue a conscientious and religious life (5.1).
America was pretty religious (Christian) in 1925. Here, Coolidge seems to equate religion with civilization and enlightenment. Take that, Christopher Hitchens and Sam Harris.
Quote #2
Peace will come when there is realization that only under a reign of law, based on righteousness and supported by the religious conviction of the brotherhood of man, can there be any hope of a complete and satisfying life. Parchment will fail, the sword will fail, it is only the spiritual nature of man that can be triumphant (11.13-14).
Shmoop's extremely educated guess is that Coolidge borrowed a turn of phrase from the prophet Zechariah, "not by might and not by power, but by my spirit alone." Ultimately, the fate of the world rests on a spiritual base; worldly stuff is not going to get us where we need to be.
Quote #3
The fundamental precept of liberty is toleration. We cannot permit any inquisition either within or without the law or apply any religious test to the holding of office. The mind of America must be forever free (24.7-9).
Because of the influx of immigrants from southern Europe (mostly Catholic) and eastern Europe (lots of whom were Jews), many Americans were worried about a challenge to their own traditions. Coolidge is making a pretty gutsy statement here about religious tolerance. Even though it was written into the Constitution that you couldn't have religious restrictions on holding office, he wanted to emphasize the point. It would be another 35 years until there was a Catholic president (Kennedy), and even then lots of people were afraid that he'd take his marching orders from the Pope.