I read a really good history of the Southern Baptist Convention, a couple of years ago (sadly, I forget both author and title) that documented the conscious decision by which the national leadership of the SBC, during the Reconstruction, made a conscious decision to be the voice of moral authority on the Confederate revisionist side, to embrace and defend the religious and social complaints of the former slave-holding class in the old Confederacy. So by the time of the rise of the Religious Right as we know it, the Southern Baptist Church had already invested nearly 100 years in raising, training, and providing volunteers for pro-segregation candidates in both political parties. After Democratic President Lyndon Johnson signed the Civil Rights Act, that put the Southern Baptist Church firmly on the Republican side.
Also in 1964, at the presidential nominating convention (per the speeches and writings of Goldwater delegate and best-sellling conspiracy theory author John Stormer), was the meeting of the Republican Anti-Communist Caucus at which the leader of the top fundamentalist seminary in America, Dallas Theological Seminary, committed to revising the curriculum to persuade all future fundamentalist ministers that fighting Communism was Christian cause number one, and to teach that it was therefore a religious duty of all Christians to support politicians from what they saw as the only reliable anti-socialist, anti-communist party, the Republicans.
In 1968, the Pope of the Catholic Church issued the encyclical Humanae Vitae, which, among other things, banned the practice of contraception or abortion. By 1968, feminism was already seen as a left-wing political cause for long enough that it was being paid lip-service by even center-left politicians in the Democratic Party, which fairly rapidly coalesced into the current situation where observant Catholics feel forced into supporting the only anti-feminist political party, the Republicans.
In the second volume of his auto-biography, Francis Schaeffer, Jr., the son of the famous evangelist (and founder of the modern fundamentalist movement) Francis Schaeffer, documents that it was his personal revulsion to the idea of legal abortion, after 1973 Roe v Wade, that persuaded him to argue his father into telling wealthy Protestant fundamentalists that opposition to abortion was the most important Christian cause, and that they needed to donate money that funded the founding of Moral Majority. Schaeffer Junior says that he approached politicians in both parties, offering them the support of Moral Majority if they would denounce legal abortion, making the argument to Democrats that the traditional Catholic origins of organized labor and their traditional embrace of government regulation made anti-abortion a Democratic cause, only to find himself out-maneuvered by feminists on the platform committees and organizing committees. So, he says, he had no choice but, as their lead fund-raiser, to encourage early Moral Majority leaders to embrace Republicans, and their embrace of traditional rural values (see neo-Confederacy, above), as the only hope of seeing legal abortion overturned. (A decision he now says he regrets, but feels that the feminists left him with no alternative.)
(*Post-1964, the Southern Baptist Church embraced the Republican Party for segregationist reasons; post-1973, Moral Majority and the Catholic bishops both embraced the Republican Party for anti-feminism reasons.)