What about Greek, the family of the Slavic languages like Polish, Czech, Slovak, Slovenian, Kroatian, the family of Finno-Ugric languages Finnish, Estonian and Hungarian, the isolated language Basque and finally the big share of the Germanic languages: English, German, Dutch, Swedish and Danish?

If we want to define language as a common value for the European Countries, then it might be some Indo-European proto-language (even then it's not clear where Basque comes from).

I agree that Latin had a huge influence in most of the European languages, but this mostly through the Roman Empire and later through the Catholic Church.