Obviously the use of "west" is vague and imprecise. However, the way I interpret it as referenced here, it is the advanced representative democracies that share the values of the enlightenment (western civilization). As such, I would include Japan, and perhaps even S. Korea, but probably not the south america ruled by despots.
As for whether it is dying, obviously from a demographics point of view, it is shrinking since most western affluent countries are not reproducing at a rate fast enough to replace themselves. Whether that will change who knows.
In the US the population is not shrinking, in fact it enjoys a healthy growth. The reason for this is immigration (both legal and illegal). Europe's population is shrinking somewhat, however, they are experiencing a surge of immigration which will stabilize it.
The difference between the US and European experience is that most of the new immigrants of the US become assimilated in a fairly short period of time, thus becoming "westernized". In Europe this is not as prevalent. Because much of their new immigrants come from cultures (muslims) that have a stronger cultural identity, and because the multicultural mindset of europeans doesn't insist on assimilation (in fact they encourage maintaining separate identities) - there is less assimilation and westernization.
Since the immigrant population is growing much faster than that of the native european, if immigration and assimilation policies remain as they are, it is a demographic certainty that the muslims will eventually be a majority in Europe. At that point they will establish laws that are more in tune with their cultural and religious beliefs.
So what would cause the "west" to die is not the decreasing population of the "european" stock, but their failure to force the cultural assimilation of the new immigrants.
This is because, the "west" is a culture, not an ethnic group.
In any case - it seems as the third world also industrializes they will also shrink population wise.
Let us not forget the great Malthusian error.