Argentina is the country whose citizenship you cannot legally lose (though dual citizenship is permitted), and they have mandatory* voting. So the Pope is still a citizen of Argentina and did vote or faced charges. I don't think they allow exceptions.
Correction, voting is mandatory only for people who live in Argentina, if you live on another country voting is optional. Source: I'm an Argentinian who lives in another country.
It is possible to vote while outside the country by going to designated embassies but if you live outside of the country or are further away than 500km from your legal residence the day of the vote you are not obligated to vote. Also, a lot of people (above 20% on the last election) that should vote don't despite it being mandatory since that law is pretty much never enforced in practice.
Source: I'm Argentinean.
This is fascinating. Assuming he follows the law of the land per the Bible (per my minimal understanding of the Bible...) he would be sinning to note vote. So the pope most likely voted by absentee ballot in Argentina, but also likely has diplomatic immunity in Argentina, and therefore could not be prosecuted for failure to vote.
I don't think you are eligible for diplomatic immunity if you are a citizen. He might choose not to extradite himself but he plans a visit in 2024.
I think he either voted for Sérgio Massa, whom he supported publicly, or symbolically cast a blank ballot. The other options are to declare that he was ill or 500 km from the nearest polling place, or pay a 50-500 peso fine. None of these are off-limits for somebody with a good diplomatic position and a large amount of staff.
What the international law cares about is "sovereign states" or "sovereign subjects of international law" not countries which is a much more informal term. Sovereign states technically don't even need a territory - 122 states have official diplomatic relations with the Sovereign Military Order of Malta (not to be confused with the Republic of Malta) which has had no territory since 1799.
Pope is technically head of state or another country so it is probably bad to vote in any election just from the perspective of looks. The pope should be above politics.