Yes. Indeed most of the religions practiced in America preach, above all other things that you think matter, putting your fellow humans first despite their background, race, social status, etc. I know Catholicism (despite some of its institutionally conservative warts) explicitly preaches tolerance, the prerequisite for social diversity.
What I've observed in America of late is a lack of religions maintaining social relevance and so increasingly becoming scapegoated by the crazies that hold out. I also consider scientific method to be an unconventional "religion" insofar as it's proliferated enough to have crazies of its own telling people how to live their lives and making dogmatic claims it itself cannot even support (your god isn't real we haven't observed it yet--would sound petty and absurd were the same argument made to counter theorized atomic particles).
It's hard because religions need to take the more difficult moral and ethical stances that no (faction balancing federated democractic) government should take, and like you mention with abortion some of those end up leaking out into the legal system.
Marriage is another thing that's danced the line but I think it falls on the other side: people let it leak into laws, or more accurately it existed in laws as the status quo, when it doesn't belong there. Two gay peole who are truly e.g. Catholic can happily live together and love each other. If they believe the Catholic criteria and definition of marriage they do not seek it. Yet they are fully allowed to openly share their love in any way the church allows for any other unmarried couple. Sexual intercourse, a beautiful enjoyable act in the church's eye, remains for the purpose of procreation and is respected not as a lustful act of pleasure, but as an intensely spiritual experience under the pretext of two married humans creating a child. In practice and due to it being socially progressive and now very safe (old "status quo thought" of abstinence is a good way to prevent spread of disease), of course people have lots of extramarital sex. But those people are not to be shunned, or excluded in most Christian sects. Extramarital sex is no more grave a sin if it between two members of the same sex as were it between a happily engaged heterosexual couple. So marriage is a religious concept that should mostly remain religious. Legally as a society it's probably in our interest to subsidize couples or parties living together (sharing space and resources) and then further subsidize those parties who choose to raise children (of their own or by way of adoption). But man it kills me how many people on both sides brought religion into what was largely a social topic during the whole marriage equality push despite the fact that I agree with the outcome.
So what is the problem? Globalization exposes fundamental radicals easier? Perhaps, but I think in America it's our ingrained social willingness to yield to "prosperous" economic policy that ends up causing the economy to ehm trump human rights. IMO it's a slow trickle of those decisions that has caused us to regress. Why is healthcare expensive, because it must prove its value economically. We don't value health enough (or we value the healthcare industry too much) to make sacrifices required to legally mandate socialized care (we would kill a huge industry, and then it's just a huge expense the government incurs that didn't exist before, lose lose). It's really that simple albeit no less unfortunate, and I personally have the health care industry to thank for most of what I've been given in life. A true doctor will take care of all patients who stumble into the ER whether they get paid market rate or not, it's part of their ethical code. So in that sense we have some sort of basic level of universal healthcare. We just haven't all agreed whether we should pay for your weed and contraception and experimental procedures and more reasonably your preventative medicine (although the business model discourages prevention while selecting for continued consistent doctor visits—another problem) too. And we certainly have failed to convey to the less fortunate an attitude of openness and acceptance and assure them that they will be taken care of to a reasonable extent by our doctors and hospitals. Insurance companies with convoluted policies and systems designed to minimize their payouts certainly don't help paint that picture.
The same thing has happened to education. Both "industries are seen as businesses and expected to run like one (make money). As long as the peripheral participants in these industries (hospital/school executive staff, insurance/loan agencies, etc.) keep demanding we run things like a business, and as long as we as Americans are okay condoning said behavior directly or by proxy of hyper economic political focus, nothing will change. Call me crazy but since when is caring for the health of your neighbor or raising the next generation to be better than the current iteration about making money? Sadly in America, it is.