There would be a western civilization, just one absent of Christianity. It would LIKELY have a strong Graeco-Roman heritage. None of the so-called barbarians wanted to destroy Roman civilization. They either wanted to preserve elements of it for their own purposes or adopt it wholesale.
Something would emerge - but it would be something very, very different from what actually developed as Western Civilization.
I agree completely that the Germanic tribes - Goths especially - wanted to preserve Roman civilization, or much of it, at any rate. (The Huns are a different story.) But any attempted fusion between Germanic and Roman culture would be more difficult - not impossible, but more difficult - without the mediation offered by Christianity. Assuming a collapse of the Western Empire in the 4th or 5th century, there'd also be less survivals of Roman and Greek knowledge and literature to sustain such a fusion. There would also be less in the way of a wider cultural or civilizational identity - unless some other equally potent faith were to take Christianity's place in this timeline.