Culturally yes, but it depends on context. As mentioned in https://en.wikipedia.org/wiki/Western_world , during the Cold War and after, "Western World" was roughly analogous with "First world".

That is to say, (again, roughly) - "modern, stable, (primarily) capitalist, democracies aligned with Western Europe/USA".

So, in the context of geopolitics and political ideologies many people still consider it to include Japan and South Korea. Yours truly included.

Not that I care as much about the "aligned with" part anymore, but certainly the "modern, stable, democracy" part.