While I believe that conservative Christians are the main source of this disrespect in public discourse, I figure that I should try to figure out where they may feel disrespected by the people whom they disagree with. Unfortunately, many public figures make constant spurious claims that Christians are the victims of frequent disrespect, yet the examples that they cite (e.g. "The War on Christmas") are typically nothing more than non-Christians challenging the cultural dominance of Christianity.
Here are the issues where I expect that conservative Christians (really, social conservatives more generally) may feel that their sense of morality is being disparaged, or that they are being prohibited from engaging in important aspects of their desired lifestyle.
- Animal rights: There's nothing worse than being called a murderer. Animal rights activists can get pretty extreme in their rhetoric. Luckily, the absolutists have no real political or social influence. While conservatives may call these people "leftists", in fact they are just as likely to target their fellow "leftists" as anyone else. In fact, one of their favorite targets is another stereotypical "leftist" group -- biologists engaged in animal research. Another favorite target of animal rightists is the hedonistic, fur-clad cosmopolitan elite who the cultural conservatives often rail against. Regular Americans eating cheeseburgers almost never receive any attention from these people. The small-town family farm of conservative ideal is typically viewed as being a pretty humane system for producing meat. Vegetarians are definitely more common on "the left" than "the right", but they are typically pretty tolerant of us carnivores, and there isn't any reason that a Christian must eat meat (is there?). The only real conflict between traditionalists and vegetarians arises over hunting, but vegetarians will typically cede the issue if the hunters make the case that it is an important part of non-elite culture and livelihood (but British fox hunting doesn't get a pass).
- Gun prohibition: As with hunting, some extremists will attribute base motives to people who want to own guns, such as bloodlust or a desire to dominate others (frankly, many Republicans have done a lot to connect these ideas in public perception). However, most calls for restricting gun ownership are framed as utilitarian anti-crime measures. As with meat-eating, gun ownership doesn't have anything to do with Christianity, though guns do play a role in some traditional lifestyles. The prospect of broad gun prohibition is politically plausible, but remote. If conservatives are concerned about losing their guns, that risk is nothing compared to the prohibitions that they are imposing on others.
- Multiculturalism/Libertinism: Conservatives are often depicted as "hateful" or "dictatorial" for holding others to certain standards. This is often a reasonable response to attempts by conservatives to impose their cultural preferences on others. However, this hostile response is not justified if the conservative is trying to persuade others to change their lifestyle in the absence of coercion. But unless we have a culturally libertarian state (including unlimited immigration), non-conservatives will be justifiably suspicious that any advocacy for cultural uniformity is just a prelude to violence.
- Abortion: Opposition to abortion is often ascribed to an attitude that women are subordinate to men. This accusation is probably made to quickly in many cases, but I cannot say that it is unprovoked; if you want to dictate major life decisions to women, be prepared to be called "sexist". While not all opponents of abortion are sexist, there is good evidence than a good portion of them are, and America traditionally was sexist, so a traditionalist should not be too shocked by this accusation.
Aside from partisan posturing and attacks on politicians, most disrespect in public discourse originates from conservatives.