The comparison to many Arab states is actually kind of scary: the idea of America going down the horrible road of Iran, by which a secular nation, due to the moral bankruptcy of its leader, is transformed into an unhinged theocracy.
The good news is, unlike Iran, America had regional governments that still remained true to the Constitution.
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About whether or not faith will still have a place in America, well, lets make some comparisons to Germany.
Germany was for a long time a deeply nationalistic society. Many, many Germans believing in crushing communism and taking back Polish lands. These values were considered pretty normal, even though people like Ferdinand Foch realized just how dangerous and destructive they could be.
Yes, not every nationalist supported the Nazis. Even people like Paul von Hindenburg who did end up working with the Nazis still found Hitler to be utterly repugnant as a person. But like a cancer cell, Hitler's malignance was not easily identifiable to the German body, and soon he turned Germany from merely a nationalistic nation into a death factory.
Thus, post-war Germany would forever be a nation that rejected hardline nationalism. Even ex-Nazis who escaped the noose would not form new political movements designed to revive German ambitions.
America's relationship with religious authority is...mixed.
Yes, there were many, many people who were guided toward good causes by their faith. William Lloyd Garrison and the Quakers used religion to push for the end of slavery. MLK and many civil rights activists used religion to push the ideal of a color-blind America. Solomon Northup was also a man steered by deep religious convictions, which kept him going in his period of enslavement. ITTL, Elvis had turned away from his dangerously hedonistic lifestyle because of his faith in God.
But religion in America OTL has some undeniably pernicious influences, predating Rumsfeld. Many religious figures have sought to use their faith to justify social control. Joseph Breen, a deeply Catholic man, used his influence to become a Hollywood censor. Because of him, references to fascism were absent from movies in the 1930s, when they could've been used to make people aware of the threat of the Nazis.
Other religious groups have long endorsed racism and segregation, or even justified it using the Bible. Like the Curse of Ham and other Bible passages cherry-picked for specific causes. Many, many Southern Baptists could preach love, while ignoring the vicious discrimination and mistreatment of their African American neighbors.
Others believed that public schools should be essentially outlets for religious belief, pushing for prayers in public schools, and demanding that books teaching evolution should be banned.
Other religious have taken up policies that were outright reactionary. OTL, Phyllis Schlafly eviscerated the ERA movement with appeals to Christian motherhood. Many preachers like Jerry Falwell and Pat Robertson happily endorsed homophobia and misogyny as "Christian values".
Other religious figures are often defenders of the status quo, seeking to use their faith to defend incredibly toxic figures. Instead of fighting the injustices of society and the exploitation of the poor, many other religious figures, including the above, endorsed wealthy and powerful people, and convinced millions of Americans their biggest problem is sex in movies and comic books.
TTL the endorsement of corrupt officials is especially notable. When Rumsfeld was taking a wrecking ball to the Constitution and workers' rights, the Christian Values Party, a party built directly from religious fervor, happily jumped on the Rumsfeld bandwagon, endorsing his unhinged and vile policies for their own ends., whether religious or temporal. The CVs used their base to push for the 28th amendment, a document so odious and contrary to American republican ideals, its existence itself should be an impeachable offense.
When the Pope spoke out against rapacious greed, the fucking Vice President, Jack Edwards, had the nerve to attack the Pope and his religious beliefs.
Jeremiah Denton, an otherwise decent man, saw little problem with what Rumsfeld was doing, and willingly joined his cabinet. It was only when Rumsfeld began treating loyal soldiers like wounded horse that Denton began to oppose Rumsfeld. But by the time Denton came around, it was too late for Denton to make a difference.
Even the principle of Christian charity was horribly corrupted, as the CVs happily exploited the impoverished American people to get their votes in exchange for some aid.
Religious authority aided Rumsfeld as he tore the US and its ideals to shreds. And when religious authority gained power, they destroyed what little chances of America remaining a world power by plunging into a violent civil war.
And the damage done to America-nuclear strikes, chemical weapons, environmental damage, mass murder-are things that Elvis and his followers can't fix. No matter how hard they try, they can't resurrect the dead, or rebuild the country. That requires the help of outside nations that are mostly secular.
Christian ideals ITTL will never, ever hold the same power and influence they once held. And in spite of the well-meaning religious individuals who resist Rumsfeldia, their help will be dwarfed by the horrible, horrible actions of people like Douglas Coe. An entire generation of young Americans will rightly blame Christian fanaticism for their hardship and the death of their families.
This makes me wonder if Christianity will be seen as the biggest threat to global peace in later years, maybe even taking the place of Islam? Or maybe, with the events both in American and the Middle East, this'll spawn a whole new generation of militant atheists who consider religion the cause behind all of this. I could see whole armies of anti-religious extremists destroying all religious institutions and artifacts they can find. Maybe this could even lead to people seeing Christianity on par with Communism in OTL. Like a Jesus Scare, where people around the world fear that anything christian is the easy road to tyranny and violence.