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Legally, it is like a private court. From the point of view of someone who is not a member, it's the same thing.The US is not part of the Italian legal system, and yet when CIA officials were in Italy, and committed crimes in Italy, and were convicted by an Italian court for those crimes in absentia, is anyone questioning that?
* https://www.nytimes.com/2009/11/05/world/europe/05italy.html
* https://en.wikipedia.org/wiki/Abu_Omar_case
The crimes were committed in a country that gave jurisdiction to the court.
> The majority of UN member states are not democracies. If all of these non-democracies would set up a court, with the legitimacy of 'the majority of UN members', would they automatically have the right to prosecute the other countries?
If the prosecuted countries/people committed crimes in the jurisdictions of those non-democracies, yes it would have the right.
The crimes were allegedly occurred in Afghanistan, and the ICC has jurisdiction in Afghanistan because Afghanistan gave it.
> The ICC lacks universal territorial jurisdiction, and may only investigate and prosecute crimes committed within member states, crimes committed by nationals of member states, or crimes in situations referred to the Court by the United Nations Security Council.
* https://en.wikipedia.org/wiki/International_Criminal_Court
If the alleged crimes happened in Somalia, which is a non-member / non-signatory (AFAICT), then the ICC would have no jurisdiction.
* https://en.wikipedia.org/wiki/States_parties_to_the_Rome_Sta...
