Why didn't Greek culture "stick" in the East the way it did in the West?

I think I see what you mean. And my answer would be: it did, to my view much more so than in the West. If you think of the various tropes of orientalism, the crowded covered suqq/bazaar, the domed mosques, the Turkish baths, the flatbread and more these stem from Roman (and by extension Greek) culture. Or more accurately, they are part of a general eastern Mediterranean culture which the Greeks are the most famous contributors to. An illustration I like to give of the differences between Western and Eastern cultural continuity with the classical world is each one's religious structures: in the West, the Catholic church is based off the design of the Roman basilica, which was a sort of all purpose public space. In the east on the other hand, church and later mosque design is clearly descended from the trends in later Roman temples--you can draw a straight line from the Pantheon to the Selimiye in a way you can't with any Western churches.

Even if we set aside these sorts of everyday forms of cultural continuity, Greco-Roman literary culture survived in a far stronger form in the East than in the West, supported by the continuation of the basic form of urban life. Medieval scholastic philosophers, after all, received their knowledge, and to an extent understanding, of classical thought from Arabic sources.

The better question is why the West has so wholeheartedly laid claim to the classical tradition in a way that modern Arabic culture does not.