Yours is a very interesting and intriguing proposition. Having thought about it a little, I came to somewhat different conclusions, however.
to begin, we cannot talk about more than Western civilization as one entity in political-administrative terms. Fiske, one of the most deep and prolific historians of the XIX century, advanced a view that history has witnessed three ideas all of at national organization: he called them Oriental, Roman, and English. In brief, under the Oriental scheme, that conqueror enslaves the conquered. The Romans, in contrast, have made those whom they conqueror equal and elevated them to the administration of the Empire. The English idea, which he emphasizes as essentially Teutonic because the team from the German tribes, is the idea of representation in government. This idea took hold on the Isles when the English conquered them and was subsequently strengthened by the English-speaking peoples, most notably, of course, on the American continent. The degree to which that idea is accepted on the European continent is smaller, and this, I believe, is the fundamental reason (besides the cultural ones) why we and the Europeans come to odds in international affairs. In other words, we do not react to the world changing events in the same manner; there is no common reaction to them on the part of the Western civilization.
The second observation is regarding they layer of Christianity that you so correctly observe to be present in the Western civilization. I think we speak here about a layer only because Europe is largely post-Christian, as they are fond of referring to their continent. Since the middle of the XIX century, the socialist ideas have penetrated the European psyche. In international affairs, too, the socialist ideology for other than that Christian beliefs are a driving factor. When we peel their layer of Christianity, therefore, we find not the Roman view of the world, as you suggest, but the socialist mentality.
Islam fails to understand the Western civilization to its peril, you say? This well may be, but given its largely socialist content, Western civilization fails to understand itself as well. That is why my concerns are equal to those that the Europeans must have had in the XVI century, when Islam was at the gates of Vienna.
This can get carried to extremes, for example the idea that killing in self-defense is wrong because the killer has just as much right to life as the victim (I don't subscribe to this mentality)
Underneath this layer, however, is the attitude that one's own self, family, group is more worthy of life than an attacker, and (in the extreme) an attacking group may be exterminated, if necessary, to preserve one's own. This is what I'm talking about when I speak of "the Roman foundation" -- the will to survive that can unleash extreme savagery when the layers of charity, consideration, and politeness are stripped away
In order to maintain a civilization, within the civilization it's necessary to maintain certain concepts, to limit the extent of conflict so that the civilization does not rip itself apart. When survival demands that these concepts are removed, the result can be very ugly