I know citing Wikipedia is some unholy internet sin, but as I can't be arsed to write a whole Ethnographic study and linking relevant readin material I recommend this...
Norther and Western Europe also absorbed a lot of these things across Europe without the necessary presence of actual Muslims.
Also the denial of cultural influences of the Ottomans in the Balkans is just Nationalistic wishful thinking. Turkish influences are everywhere, in the cuisine, language, arts, clothing, music, dances, literature.
What I was trying to say is that while differences exist and sometimes are very deep, the exchange of population and culture and customs always existed within Europe and into Europe. Europe can't afford to isolate itself as much as some wish. We have an aging population and a dependance on imported goods and a dependance of exports. If we want goods, knowledge and information to flow, we need to accept the people that come with it.
What we really need is to figure out how to adapt to these new times and how to control and absorb immigrant populations while maintaining the defineing charactersitics of our European culture.