In Western culture, we tend to say God instead of saying the Christian God. Why?
The answer lies in your phrasing: “Western culture” is the collection of social/economic conditions from the overwhelming influence of the spread of Christianity in the Dark and Middle Ages through what is now known as Europe. Christianity, an offshoot of the Judaic religion, was/is a religion that maintains the Jesus of Nazareth was the Messiah, the savior that the Jews were promised and were waiting for – contemporary Jews deny that he was the Messiah, and they are still waiting. As Christianity spread through the continent, in the form of churches, monasteries, battles, etc., and eventually as the politically established religion, the other religions – Greek, Indian, Islam, etc. – lost power and eventually were outside the culture of European “western” peoples. Several incidents (such as the barbarian invasions, Druidism, the Crusades, etc.) threatened the dominance of Christianity for short whiles, but did not prevail. Today in western countries (those settled by European explorers), the percentage of Christians is large enough that in ordinary conversation the word “God” is a short-cut for Christian God. In multi-cultural conversations, the word is modified into terms such as “Christian God”, “Muslim God” (Allah), Native American God(s), etc.