Well is it?It seems to me that a continuing cultural thread runs from Mesopotamia passing through Europe and finally arriving in North America.Or Greece and Rome are a breaking point with "Oriental" cultures?
This isn't the sort of question you find citations for, but the essence of Western Civ. starts with the Greeks. Much of Greek history lies in its interactions with the Persians and other denizens of empires and kingdoms in Asia Minor, the Levant, and Mesopotamia, so going back over the history of these regions is of interest. The Civilizations in Egypt are also if interest for the same reasons. And in the end, Alexander conquered the Mideast and Persia and introduced a dose of Greekness into the mix.
So Sumer is of interest to Western Civ since it sets the stage for the East/West interaction that the Greeks had. Other states farther off that did not interact with the Greeks are not considered much - India, the Far East, Africa. These are studied in their own courses.
"Western Civilization" is a very loosely defined expression. It can be more or less whatever you want it to mean. Usually, however, when we talk of "Western", we implicitly place ourselves in a modern context, where Western civilization really means "industrialized countries with a large middle-class and partaking to a common set of values which includes consumption of hamburgers". Roughly speaking, this encompasses USA, Canada, the richer half of Europe, and Japan. Depending on who is speaking, considerable variation may occur in this list, but the important point is that it is always anchored in recent times. Sumer is 4000 years old.
There is no clear-cut separation of "civilizations" with well-defined boundaries, except in the extreme cases of 100% geographical separation; e.g. pre-columbian societies in Americas had been separate from eurasian societies since before the invention of cities, hence before the notion of "civilization" can actually be defined. On the other hand, societies in India have incorporated elements from Greece (Greco-Buddhist art is a blatant symptom of this), so if you want to follow your "continuum" then "Western Civilization" would have to include India as well; and then China, if Buddhism is promoted to "Western" status. The notion of "Western" vs "Oriental" then becomes meaningless.
At best, we can try to talk about "ethnies" and trace them through languages; languages spoken by most people (but not all) in Europe and North America come from the Indo-European family. That family also includes the majority languages spoken in "oriental" countries nowadays (Iran, Pakistan, India...); but Sumerian is not indo-european. In that (very loose) sense, Sumer should be considered as "less western" than modern Iran. But really, the notion of "Western Civilization" is not well-defined.