The Catholic Church — a politically and ethnically sprawling institution — has no natural home on the American ideological spectrum. Neither major party combines moral conservatism with a passion for social justice. So Catholic leaders have often challenged Democrats to be more pro-life and Republicans to be more concerned about immigrants and the poor.

But President Obama’s first term was a period of unexpected aggression against the rights of religious institutions. His Justice Department, in the Hosanna-Tabor case, argued against the existence of any “ministerial exception” to employment rules. Obama tried to mandate that Catholic schools, hospitals and charities offer insurance coverage for contraceptives and abortifacients. His revised policy still asserts a federal power to declare some religious institutions secular in purpose, reducing them to second-rate status under the First Amendment.

On top of this, Obama ran a stridently pro-abortion reelection campaign, seeking culture-war advantage on an issue he seldom mentioned four years ago.

The Catholic hierarchy and more traditional Catholic laymen reacted as you’d expect. Bishops issued pastoral letters in defense of religious liberty. Conservative and pro-life groups organized in battleground states.

The result? According to the first cut of exit-poll analysis by the Pew Research Center, Obama’s support among white Catholics fell to 40 percent — seven points lower than four years ago. It was one of the largest swings of any portion of the electorate. John Green of the University of Akron argues that the religious liberty issue came to “encapsulate other concerns such as abortion and marriage” among many regular mass attendees.

In a close election, this reaction might have made all the difference. But the election wasn’t particularly close. And the trend among white Catholics was partially offset by Latino Catholics moving in the opposite direction for reasons unrelated to abortion or religious freedom. (Obama gained three points among Hispanic Catholics and took three-quarters of their votes.) In the end, Obama won the total Catholic vote by a small margin.

This result reveals a tension at the heart of the Republican coalition. The portion of that coalition that is pushing away Latino Catholics is making the political work of conservative Catholics far more difficult.

Catholics have a historical advantage in understanding the imperative of inclusion in modern politics. They belong, after all, to an institution that has been multicultural since Peter first set foot in Rome. But white evangelicals are now getting their own education in coalition politics. They gave Mitt Romney a remarkable 79 percent of their vote — the same share that George W. Bush received in 2004 — while comprising a larger percentage of the electorate than they did 2004. But their energy and loyalty were rendered irrelevant — washed away — by GOP failures among other groups.

“Rather than a repudiation of cultural conservatism,” concludes Green, “this was an election in which cultural conservatives did everything they could, but the party fell short.”

In the long run, social conservatives will have serious trouble exerting influence unless they are allied with rising ethnic populations, which tend toward conservative social views. But social conservatives are now in a toxic alliance with political forces — the wall-builders and advocates of self-deportation — that are actively alienating rising ethnic populations.

Evangelicals and conservative Catholics — some of the most loyal members of the Republican coalition — have a direct political interest in making that coalition more inclusive. Hispanic outreach alone is not sufficient. Romney’s largest problem — picking from the smorgasbord is a challenge — was probably his underperformance among white working-class voters. But given America’s demographic direction, the overwhelming loss of Hispanic votes will gradually complicate the Republican political task to the point of impossibility. Unless this problem is solved, the GOP will remain on a long, downward slope toward irrelevance.

Outreach is not done in a single awkward lunge. It will involve more than endorsing comprehensive immigration legislation, though that is necessary. Hispanic voters have a series of concerns typical of a poorer but economically mobile community: working schools, college access, healthcare, a working safety net. Republicans will need to offer policy alternatives on these issues — defining an active, market-oriented role for government.

Perhaps the greatest Republican need is to embrace and demonstrate some other sound Catholic teachings: a commitment to the common good and a particular concern for the poor and vulnerable. This might appeal to Hispanics — and others.