I'm partially sympathetic to bashmelek's point of view. So much of the history and culture of Western civilization is actually rooted in the eastern Mediterranean. In the old world all writing can be traced back to Mesopotamia/Egypt and China. A surprising number of areas still use "ABCs" to refer to their alphabet. Much more than half of the old world by geography has a deeply connected written scholarly history. During Antiquity Greek culture itself extended much further east than west. Iconic Hindu and Buddhist art, particularly statuary, is Greek in origin. The center of Christianity was still east of Europe, or at least much further east than Rome, into the Middle Ages. There's a fascinating account by a circa 12th century papal emissary sent to the capital of the Khan Empire in Mongolia on a diplomatic and proselytizing mission. When reaching the Khan's court he found an entire
native Christian contingent, and rather than listen to the emissary's preaching (Christianity wasn't new to him, and he wasn't inclined to take it up, though a sizable fraction of others had), Güyük Khan had him debate Christian theology with the existing Christian council in his court, partly for his own entertainment. The existence of these Christians caught the emissary by surprise, because the history of Christianity in the East had been forgotten by the Europeans. A few decades later, the Mongol Christian wife of Hulagu Khan was instrumental in securing the safety of Christians during the Sack of Baghdad (and IIRC she even had commissioned, or tried to have commissioned, the construction of a new Christian cathedral in Baghdad. But I might be thinking of another, much later Mongol Christian empress in China.)
All of which is to say, a huge portion of the old world can claim "western civilization" as an inheritance, unless we narrowly define "western civilization" as being rooted sometime in the past millennia (and then ignore the the effect of European colonialism in bringing many other areas into the fold, some for the second time). And that's before delving into pre-history and the influence of Central Asian steppe culture, from which we get Western civilization's shared Indo-European linguistic and, arguably, even religious history (see, e.g., links between Judeo-Christian and Dharmic religions via Zoroastrianism).
Which isn't to say modern European culture has the same relationship as everywhere else touched by Mediterranean culture, or that Renaissance and Enlightenment ideals aren't important, pivotal contributions to modern "western" culture. History has attenuated, obscured, and partially erased the historical interconnectedness. But relative to, e.g., East Asian or Mesomerican culture, there's a strong argument that everybody from the Hindu Kush to Portugal, from Ethiopia to Moscow, have a deeply shared cultural history and inheritance and are rightly called "Western" in a meaningful sense.
I don't object to the existence of a western civilization, I just object to Europeans claiming it as a unique, or even uniquely formative, inheritance. When I read about Hindu nationalist claims to a third emergence of writing in the Old World, it seems they're so passionate only because there's the tacit belief that they don't have the same stake in western civilization as Europeans; that they're not otherwise equal heirs to a rich western culture. But they are, IMO.