"white" as a claim of culture or heritage, only exists in the context of white supremacy. The concept of "whiteness" was invented during the days of the slave trade as a means of establishing racial classification and hierarchy - to contrast the superior "white" race with inferior "non-white" races, which is why it traditionally excluded people like the Irish and Italians, along with the obvious African peoples traded as slaves, who unlike the former groups can never "ascend" to "whiteness" no matter how much political or cultural capital they accumulate.

In the modern day, it has lost much of its explicitly racist connotations (except, obviously, when explicitly used in the original context by racists,) but just tends to mean "a person who looks white (until proven otherwise)" or "of European descent," the former of which isn't really related to culture, and the latter being so vague as to be useless as a cultural referent.