The universe is cruel in such a way that it does not care for the continued survival of "higher, more cultured ideals", no matter what sort of egalitarian ideal we follow. In most scenarios, the breeders, the masses, simply win out.
The high point in Muslim civilization witnessed this natural event themselves, when the Mongols attacked them and destroyed much of their civilized culture up to that point. Intellectually, they never recovered from that. They were out-bred, out-fought and out-willed, by what was at that time a more energetic, expansive collective of people, and were forced to adopt cruder methods than the cultivation of science and arts, in order to survive.
So will the European people take a back-seat to the other peoples in the world, unless there is some sort of a cultural awakening, which allows the European man to appreciate what he has achieved, and encourages him to hold onto it, preferably without lashing out like a petulant child once the reality dawns upon people.
The shock of gradually turning into the minority, no matter how big on social justice a person might be, is something that 90% of the people will not be able to handle. Instincts kick in, at that point.