They don’t pay taxes to the city of Rome either, despite using our services and infrastructures: they use Rome’s sewage system but don’t pay for it, they use Rome’s garbage collection system but don’t pay for it - we are talking about private homes, not just churches. They don’t pay for the police service, streets maintenance and so on. The Vatican is s literal parasite of the city of Rome and the Italian nation as a whole. They are not an independent state, they depend on us for everything, hence they should pay taxes for the social services they use.
To be fair, this sounds like it's also on the Italian politicians and therefore the Italian public for not having done anything. Just start charging them, what are they gonna do? Either cough up or become a stinking pile of shit.
Well, Vatican city was already an Italian possession after the unification. The fascist regime gave it back to the papal state in exchange for the Church supporting Mussolini and fascism (Patti lateranensi).
Seriously, fuck the taxes. I mean, fine, tax them where you can, but that is BY FAR the least of the worries. Churches in the US are tax exempt because they are organized as non-profits and charities. You either have to change those tax laws or audit the shit out of each of them, neither of which is a quick or easy solution. 
There’s two things I think can be done given the political will/public pressure. First, do not let institutions do like the catholic church does and let an individual diocese declare bankruptcy after losing a suit. The catholic church is a holistic organization and the entirety of the church should bear the blame for what it’s representatives did. Use the income of the entire church to determine damages, and make the church responsible for paying them. Seize assets if it declares itself unable to pay. Same with the mormons and any other religious institution.
Second, send bishops and other administrators to jail for criminal conspiracy if they knew of an incident and covered it up. Set mandatory reporting requirements that disregard religious ideologies like the “seal of confession.” Treat it like the criminal conspiracy it is. In the US you can bring in the feds because they moved criminals between states to escape justice.