I wonder if you might look at your question again to see if you have asked it correctly. As stated, this question cannot be answered. The reason for this is that the Fertile Crescent is in Southwest Asia. We can see that Diamond uses them interchangeably from the fact that the entry for "Southwest Asia" in the Index says "See Fertile Crescent."
The Fertile Crescent was able to be the source of civilization because it was such a good place for farming to develop. In Chapter 8, Diamond runs through a long list of reasons why this is so. The early adoption of farming, as we are told in Chapter 4, leads societies to become powerful. Chapter 13 tells us how it can also lead to innovations such as writing. In other words, places that started farming early were the first to have civilizations that could spread. Finally, Chapter 10 tells us that culture could spread from the Fertile Crescent because Eurasia has a long east-west axis.
These facts show us why the Fertile Crescent was a major source of culture.