No, he speaks a mix of Italian and Spanish. He understands everything obviously but most of the time, he starts his answer in Italian and finishes in Spanish.
My guess is, he probably learnt his Italian "on the job", probably by hanging out with Sara Errani and maybe other Italians.
It is often said that Spaniards and Italians understand each other relatively easily, because of the similarity of the two languages, but I've always been mystified by this (I've studied both, and to be honest, they are very much alike but there are also many differences - when I see how difficult it can be for French people to understand French Canadians, who speak mostly the same language as us, I find it difficult to believe that Spaniards could understand Italian just like that).
Well, anyway, he understands it well, and knows it enough to speak a little, so maybe if I ever get to talk to David, I should speak in Italian