It is not recognised by the UN
It is recognised by the UN. It has observer State status. Only a dozen countries do not recognise it as a state.
it does not have borders
It has borders with Italy that have been agreed upon and formalised into law.
It has a gendarmerie and a military.
its relations are not of a full diplomatic nature.
It issues passports and during WWII had embassies inside its country that were recognised by Britain and Germany amongst others. Not maintaining full diplomatic status with all countries is meaningless as that changes with all countries from time to time. Nauru and Monaco are examples of small countries that dont have a lot of embassies. Israel has status and it has very very few extradition treaties with other countries.
Germany is the only country that might prove a link between when the Pope was a bishop there and allowing child molesters to escape the law, and they have already recognised Vatican City. There are many countries that would not support such a move and a lot of heads of state would not either. The only precedent is crimes against humanity, an evolution of WWII war crimes, and that wont wash in this instance.
Stephen Hawkings was grandstanding on a suggestion by a retired Australian court judge, the original meaning of which was that it MIGHT be possible for Germany to charge him, but it would never be legal to arrest him and bring him to trial. He would simply be expelled from the country, persona non grata . Certainly crimes against humanity charges would never stick and make a mockery of their intent.
The pair believe they can exploit the same legal principle used to arrest Augusto Pinochet, the late Chilean dictator, when he visited Britain in 1998. This was based on extradition treaties with spain and as an ex-head of state he did not have diplomatic immunity. The crime will be dated 1985, when the Pope was in charge of the Congregation for the Doctrine of the Faith, which deals with sex abuse cases.
The pair believe they can exploit the same legal principle used to arrest Augusto Pinochet, the late Chilean dictator, when he visited Britain in 1998. The lawyers believe they can ask the Crown Prosecution Service to initiate criminal proceedings against the Pope, launch their own civil action against him or refer his case to the International Criminal Court. Last year pro-Palestinian activists persuaded a British judge to issue an arrest warrant for Tzipi Livni, the Israeli politician, for offences allegedly committed during the 2008-09 conflict in Gaza. The warrant was withdrawn after Livni cancelled her planned trip to the UK.
As a sideline The Congregation for the Doctrine of the Faith is the name which the Office of the Inquisition changed to, but it was a name change only. It still bans books and works from the same building.