In this case, No, you do not capitalize “western”: the “west” referred to here is not a specific place or region named “West”, just vaguely “the cultures which derive from Europe”. It’s not even “the countries west of the East” - consider Australia.The “West” as the region into which the U.S. was manifestly destined to expand and the “Western States” when newsanchors are talking voting blocs are specific and different.
You only capitalize “West” and “Western” in the non-oriental sense if you're going to say something ✲IMPORTANT✲ about ✲Western Civilization✲ or ✲The Decline of the West✲.
And “western food” is perfectly understandable. I’m not sure I could define it; but it’s what I eat every day, and it definitely more like what people from Poland and Argentina and New Zealand eat than it is like what people from China and India and Nigeria eat, which I only get as an occasional treat.
Well, it appears that I have demonstrated my provincialism, and I must bow to my more cosmopolitan colleagues. I retract the answer. I may delete it in due course (if the community does not take that sad chore off my hands), or I may leave it to memorialize my error.
But me, I’m unregenerate. I am embedded in western history, I read western literature, I listen to western music. Western history is the Chisolm Trail, Western literature is Zane Grey, and Western music is “Tumbling Along With the Tumbling Tumbleweeds”.