I see this statement repeated uncritically in conservative/conservative-lite circles, and I want to get to the bottom of this. On the one hand, it's almost impossible for Christianity not to have an impact on the moral development of the west (it's been the dominant political/ideological force for hundreds of years). On the other hand, there is the argument that the Enlightenment had more of an impact on the development of Western civ, and much of what we cherish in the west (human rights, reason, liberty etc) are direct descendants from Enlightenment philosophers who were secular and/or anti-Christian. I guess much hinges of the definition of 'western civilization,' 'Judeo-Christian,' and 'values.'
You started in your last sentence by doing what I would do with such an assertion. Deconstruct what the assertion is and try to disprove their theory by pointing out that "Western Civilization" is a term that is built on preconceptions.
"What is Western Civilization?" For example, do they mean Anglo-American? Or, as often the case with these assertions; does it mean "White" culture? Furthermore, Judeo-Christian is a huge, if not massive set of different values. Do they mean Protestant values? Catholic values? Orthodox Christian Values? Why stop at Judeo-Christian? Why not Abrahamic Values (so it would then include Islam). By pushing on clarification you will push the asserter to define their "values" that they mean. Often the case is "white" values. Or they fall into the Appeal to Tradition fallacy. "We are a country founded on Christian values" is a fancy way to say "We've always done it this way." Also, what specific values are "we" supposedly founded upon? What religious values somehow was the paragon of our culture that led us forward?
However, I feel like you are close to maybe coming to a position. That maybe "Western Civilization" (whatever that means) is based on the conflict between reason (science, liberalism) and tradition (religion, value-based).
There's definitely more to say on the topic, but in regards to your comment that about the definition of 'Judeo-Christian', I've discussed the history of the term 'Judeo-Christian' at a broader scale here, while /u/kugelfang52 has a great post on the way the term was used in the American military in World War II.
I'd also point out that I've got a book on my bookshelf called The Passion Of The Western Mind: Understanding The Ideas That Have Shaped Our Modern World View by Richard Tarnas, which takes a fairly standard, conventional 'greatest hits' approach to intellectual history (that is, the kind of stuff that people are talking about when they talk about 'the values of Western civilisation'). This book starts not with Jewish or Christian thought, but instead Ancient Greek thought; Tarnas comments that:
...our way of thinking is still profoundly Greek in its underlying logic, so much so that before we can begin to understand the character of our own thought, we must first look closely at that of the Greeks.
In attempting to discuss the intellectual history of Western civilisation from Homer to Freud in 6 chapters, Tarnas eventually gets to Christianity in chapter 3.