A place is Western if the West accepts it as such.
Pretty much. We can all basically agree that the west includes:
USA
Canada
Australia
New Zealand
Scandinavia
United Kingdom
Ireland
France
Benelux
Germany
Switzerland
Leichtenstein
Austria
Iberia
Monaco
Italy
San Marino
...anywhere I forgot?
Then there are a few nations that are almost always agreed upon (Greece, Israel, Malta, etc). But I think it would be a fair way of working it out; if 4/5ths of the 'universally agreed west' regard a nation as western, it is. The problem is that the question isn't important enough for the west to put its neck out and define itself.
Plus
Vatican
Turkey
Minus
Australia
New Zealand
Israel
I would personally include Latin America, aside from certain countries such as Venezuela, Cuba (unfortunately), and Bolivia.