Title, I am curious about the Cathoilc Popes stance on war, what happened when two nations that took the Pope as their religious head fought each other ? Would the Pope try to establish peace ? Condemn one side and not the other ? And if he did, did his condemnation have any value ? Did his word change the outcome of a war ? I guess this could all be jointed into one question, how much power did the Catholic Pope have during war time ?
I can't answer for the subject of the modern history, but I will try to answer for middle ages. Firstly, we must be aware that for a long time pope was also head of state, a monarch of the papal lands and a fair share of their policy was lead by it. Papal support for the Franks have also a lot do with the Lombard threat to the Church's power and popes weren't shy of calling Franks to attack them despite the fact that the Lombards were Catholic as well. In this time Church became closely intervined with the emperors of holy roman empire and that was one of the key aspects of their policy till the break between them, which was mostly a consequence of the struggle for investiture and that divide grew stronger and stronger, because the Church wasnt too fond of the emperors claim to the part of sacral authority. That have reach the final stage with the Hohenstaufens and their struggle against the Church, and therefore popes called was against the emperor Frederic II a crusade.
In this struggle papacy clearly call other catholic monarchs, mostly France and England to their aid and came out as triumphant and sole claimant to the sacral power. Connection with the Church, secular interests and Orthodoxy of the nations tended to lead Churchs policy toward different kings and some were clearly preffered by the popes mostly because of their loyalty, in this case we can mostly say that Spain and Habsburg empire showed themself to be more willing to do the bidding of the church than more pragmatic France and Venetian republic.
Nominally, of course popes always championed peace and were up holding to the theory of just war, which is fairly complex, but when talking about practical policy, decisive factors were support for the church and Orthodoxy from the side of the countries in question.
It varies from Pope to Pope. The big example your are looking for is the Peace of Nice of 1538. Pope Paul III effectively brought peace between the Habsurg Emperor Charles V and the French King Francis I. It lasted only for two years but it was considered an extraordinary success for that period. Not all the Popes were neutrals. In fact, the predecessors of Paul III were very much involved in European wars supporting one side or the other. Paul III ,when elected, declared its neutrality and the necessity to bring peace among catholics in order to face the Protestants and the Turks.
Relazione di Soriano Antonio da Roma (1534) e dal Convegno di Nizza (1538), Relazioni Ambasciatori veneti al Senato, Eugenio Alberi