The Italian defence minister orders the attack to be aborted after which the missile veers of course and, presumably, detonates off screen. However air to air missiles are designated fire and forget which means that when fired they are no longer under anyone's control, including the pilot.
When the Italian fighter jet launches on the drone it fires what appears to be a heat seeking missile against it. Heat seekers lock on to the engine exhaust of aircraft which, usually, is quite considerable. Such a small drone, as used in the episode, would have nowhere near the signature to facilitate a lock on.
The Italian fighter shown over Rome is an F-18, which the Italian Air Force has never owned. They have however owned F-16s.
When the Italian missile is shown being launched, it's from a delta-winged Eurofighter Typhoon, which is not the F-18 fighter shown in the longer shots.