The Vatican is an absolute monarchy. Watch CGP Grey's video on it, as far as tertiary sources go, it's gold. ;)
A state must be able to defend its borders, the Vatican cannot.
Ok, so I typed up "define state", and here is the country-definition, for state.
a nation or territory considered as an organized political community under one government.
Ok, since we're saying that the Vatican is a nation, let's see what qualifies something as a nation.
The 1933 Montevideo convention defines a state as:
The state as a person of international law should possess the following qualifications: (a) a permanent population; (b) a defined territory; (c) government; and (d) capacity to enter into relations with the other states.
Where here do you see "the ability to defend its borders"?
This is no strawman; I'm addressing directly what you said, replying with how you're wrong.