Latin America is certainly considered part of "the West". Latin Americans are Christian, they are Catholic, they speak European languages. The area is known as Latin America, i.e. descended from the Western Roman Empire!
Yes, Latin America has some non-Western roots: Indian ancestry, language, and culture, African culture and ancestry (in Brazil and the West Indies) - but so does the U.S. The U.S. even includes some Polynesian (native Hawaiian) elements.
While I would agree that Latin America is "Western", Latin America's undevelopment (I'll add the Caribbean to this too considering many countries there) compared to Spain and Portugal is certainly a reason for seeing them as something different.
What's undisputable is that Latin America has far higher amounts of American Indian ancestry (Bolivia, Peru, Paraguay, where Spanish speakers are a minority in many parts), as well as African ancestry (Brazilian Northeast, for instance), well, at least in Brazil and certain parts of Colombia, Venezuela, and especially Cuba, Puerto Rico, and the Dominican Republic. Chile has Polynesians too, for the same reason the US (who has far, far more) does--imperialistic annexation of their homeland.
I don't think a lot of people could really go to some rural village in the Andes (Bolivia, perhaps), which speaks Spanish as a second language and has older individuals who can barely speak it, and say "I'm in a Western country". Western is a hard definition to pin down, yes, but I'd argue it implies the subordination of indigenous culture to the Western ideal. I think development in Latin America would cause an acceleration of that process, at least prior to the mid-20th century.
Basically, I'm arguing that much of Latin America is Western, but on the periphery of Western culture, and could be brought further into it through increased development. I think Russia is much the same way--I mean, the debate amongst Russian intellectuals in Tsarist Russia that basically went "Is Russia a Western nation" was extremely important historically. Latin America, especially the Andes and Mexico where indigenous cultures persisted far more vibrantly despite their issues, is similar.