In Europe and the Faith, Hilaire Belloc argues that European civilization, qua European, is essentially Catholic; that, in effect, what makes Europe united as Europe is that it became Christendom. As he puts it, "The Catholic alone is in possession of the tradition of Europe: he alone can see and judge in this matter." Because of this, understanding Europe and its history requires a 'Catholic conscience'; and Europe will inevitably perish as Europe if it does not return to the Catholic faith.
In Philosophy and the Crisis of European Man, Edmund Husserl argues that Europe, understood as "the unity of a spiritual life and a creative activity" is entirely non-geographical -- in fact, includes the United States and much of the British commonwealth and excludes many geographical Europeans like Gypsies -- is characterized by philosophy in a broad but standardly Western sense, i.e., the sort of endeavor that has roots in ancient Greece. Thus Europe is an unfolding of the theoretical attitude in which the Greek philosophers tried to understand the world around them; this includes not only philosophy in the narrower sense, but the sciences. Innocently, but perhaps more than just a tad ominously, he says, "Philosophy has constantly to exercise through European man its role of leadership for the whole of mankind."
According to Robert Schuman's 9 May 1950 Declaration, Europe is something that must be made by the establishment of international solidarity, and is required for its potential contribution to world peace.
In From Ethnicity to Empathy (PDF), Ash Amin argues that the current demographic facts of Europe require that it be understood in terms of empathy with the stranger.
In a July/August 2006 edition of The Liberal, George Steiner has argued that Europe is expressed in five traits: coffeehouse culture; traversable and human-scale landscape; streets and the like names after artists, writers, statesmen, and so forth; a "twofold descent from Athens and Jerusalem"; and an apprehension of a "closing chapter, of that famous Hegelian sunset, which shadowed the idea and substance of Europe even in their noon hours." He also suggests that its genius lies in its emphasis on the highly pluralistic notion of (using a phrase from William Blake) 'the holiness of the minute particular'.
And these are but a sampling. There are many Europes. And that is what one should expect, isn't it?