Fascinating hypothesis! I will ask for confirmation. In Italian (at least in modern Italian), scarp is either scarpata (sensu geologico) or a very unusual scarpa in building-jargon (synonym of barbican?).
I poked around the Slovene dictionary and discovered that škarp is there, marked as "usually pl." (i.e. škarpi) and "low vernacular" (code for "people say this at home but don't you dare use it in formal writing"), with the meaning "an old, worn-out shoe" or sometimes (dialectically), just any old shoe.
But next to it was the word škarpa, which is a perfectly respectable word that means "a wall between two different heights of land, to prevent erosion or landslide"; in other words, a scarp or (more commonly) escarpment, which my English dictionary tells me comes from the Italian word scarpa. I think this solves the mystery. When you're soaking up the sauce with you're bread, what you are doing is making a "little scarp", just like on Chained_Bear's beloved fortresses. So you're not doing scarpetta; you are making (i.e. constructing) a scarpetta.
Edit: The escarpment has been repaired. Thanks, Bil!
So I want to know about the word itself (sorry, guys, that's just the way I am). One translation tool tells me that scarpa means "shoe" in Italian. So when I sponge up the leftover sauce with my bread, am I "doing/making the little shoe"?
if your Italian date asks for permission to do scarpetta during a romantic dinner... he's not talking about playing footsie, much less it is a comment about your shoes. (Further information on Wikipedia)