Asking current Americans what's wrong with nudity is not really the best approach for a matter which is 100% historical, and as with anything to do with culture, hard to explain solely on current perceptions (because often the current perception isn't the rationale, and people do things naturally once it's internalized on a large scale). You know, and this is pretty relevant in anthropology, that often there doesn't have to be a current explanation for anything... the passage of time makes societies forget why they started doing something and there's so much in the world this applies to. But we can't really connect it to religion, at least as the main reason: shame over nudity goes as far back as Ancient Greece, because only the barbarians were naked in public. At least until the Classical era where nudity became more acceptable during public exercise. In fact, the reason women generally couldn't expose as much skin as the men in western society can be traced down to Spartans allowing both men and women to be naked and compete in nude events, which Athenians saw as barbaric since they only allowed men to be naked during those situations.
Still while this gives us some perspective over how people see nudity, it doesn't really answer why there are circunstances it's more acceptable than others... and that's more class based, I think. Catherine de Medici held court festivities with topless women and that was seen pretty casually but at the same time, it was considered more appropriate, generalizing broadly in Europe, for a poor woman to expose her breast than a noblewoman - so the queen's servants, for instance, would show their breasts but never the queen herself or her family. That's still partly true today; you have probably seen those National Geographic pictures where pictures of African villagers weren't censored, but naked American or European women were. Social standings among different populations, dig?
Also, adornment has been used as a sign of status ever since humans started wearing clothes at all. Clothes show how rich or powerful you are and they are the last thing to be taken from the destitute.
So I don't think it started as a religious thing, much less a Judeo-Christian thing. For one, look at uncontacted tribes - some wear some kind of loincloth or in some way cover the genitals. In ancient Chinese myth, it's said the Yellow Emperor invented clothes (and pretty much everything else). Obviously he didn't, but the point is that his traditional dates are around 2700 BCE so it's reasonable to think that Chinese people have been wearing clothing for at least that long. If you look at the artistic record of China, there's also pretty much no nudes. And I like said on my first paragraph, Greek nudity only became common after the Classical era. Besides, there is the practical aspect - depending on the weather of the region you live in, it might be advantageous to wear certain types of clothes. So societies which made that a tradition ingrained into their culture thrived.
Let's move on to why violence is accepted, but not nudity. A surprising amount of people say it's due to Puritan tradition, and I don't buy that, because puritan values didn't influence American culture as much as revisionist Victorian history likes to say. (I'm not American, by the way.) It's totally true that America was basically founded by puritans who were too vocal for the British elites, but America very quickly became more influenced by a lot of other different Christian denominations which certainly had more impact than the initial puritans.
And while I do believe neither nudity nor violence should be censored, unless it's in extremes, there is an important aspect to consider. People (specifically teens) are more likely to engage in impulsive sex acts than they are in impulsive acts of violence. Normal children see violence and they know it's not ok, they know it's fantasy. But they see promiscuity, and the lines aren't so clear. What's the problem with showing sexuality? By itself, and by basic standards, none. But in any place, but particularly in America due to how it developed so much so fast, "another form of control". Teaching social freedom, especially in regards to sex, means that the other social norms will be called into question. Some people may equal this to propaganda and control we'd see in a totalitarian regime... it's not, and I'd smack anybody who made that comparison, but the reality is that this is what the streamlining of cultures leads to (in this situation). Violence, on the other hand, is not only more easily understood as a negative thing, it can also be teamed with righteous fury or whatever other characteristic you tie it to. It's more impersonal. Watching violence on television doesn't make you violent, but watching porn on TV, conditioned (and natural) behavior may get you randy.
For the record, I don't believe nudity is pornographic. I do believe media shouldn't be ashamed to show naked people unless it's deliberately used for sex appeal. If you show a bunch of guys in a gym shower room, there's no particular reason to hide anything... no particular reason to focus on showing anything either, but if it's appropriate, you shouldn't need to change your production because of one detail. Especially when it goes against logic - how do you explain a woman wearing a bra right after a sex scene? It's not very logical, and that's a case it wouldn't be a problem to show nudity. I saw Ben Affleck's schlong in Gone Girl and it wasn't pornographic or erotic at all, it would have simply been awkward to position the camera in ways that didn't show it considering the nature of the scene. So that's an appropriate showing of nudity in a movie. But my point was to show why we generally don't see much nudity in media, and although you'd need further research for more specific developments, it seems simple enough to understand what led to this perception in global culture.