Vatican is not internationally recognized state and it is not a nation state anyway. That is the reason that Hitchens have proposed that the pope is arrested if he arrives in UK: he is not a head of state and has no real diplomatic immunity.
Technically, you're correct. And, the Vatican is not the government. It's the Holy See. The Pope is the head of both, and according to the U.S. State Department, "As the 'central government' of the Catholic Church, the Holy See has a legal status that allows it to enter into treaties as the juridical equal of a state and to send and receive diplomatic representatives. The Holy See has formal diplomatic relations with 178 nations (among which it includes the Order of Malta), including the United States and some predominantly Muslim countries.