Napoleon Bonaparte (1769–1821) is usually thought to be the inspiration making France the answer to your question.

However, Italy also claims it as Neapolitan ice cream relating it to Naples Italy.

We know Italy had ice cream before the French so my belief is that it originated in Italy.

This ice cream is often called by both names Napoleon and Neapolitan
which brings confusion for its origins.

Edit: The origins of ice cream can be traced back to at least the 4th century B.C. Early references include the Roman emperor Nero (A.D. 37-68) who ordered ice to be brought from the mountains and combined with fruit toppings, and King Tang (A.D. 618-97) of Shang, China who had a method of creating ice and milk concoctions. Ice cream was likely brought from China back to Europe. Over time, recipes for ices, sherbets, and milk ices evolved and served in the fashionable Italian and French royal courts.

http://inventors.about.com/.../ice_cream.htm

I believe no one knows for sure..thanks for the comments :)