To me, the most important defining feature of western culture is the corpus of literature, art and music, sometimes called the Canon, or the Great Books. Mostly Dead White Guys, I'm afraid.
Of course, other cultures have their own collections of lit, so it's not the existence of literature in general that define the west, but this specific example.
Every culture, by definition, has social norms, ethical values and traditional customs that differentiate it from others. This is hardly surprising. But it's a bit weird to claim that grammar (just for one example) is a unique contribution of Graeco-Roman culture to human history. A brief study of Sanskrit knocks that one on the head- the Greeks neither invented grammar nor were its greatest exemplars (although we use their word for it).
But of course Greek/Roman grammar had great influence on the world's second most successful language, the otherwise largely germanic English. And the Roman alphabet (though totally semitic in origin) is probably the world's most prevalent (tho I don't have any figures to back this up). Of course the reasons why these things are true are seriously open to debate and might not be anything to be proud of.