I read and hear arguments about defending The Catholic Faith against Protestants and sometimes Jews but I've never heard a discussion between a Catholic and an Orthodox Christian about which church is the church founded by Jesus Christ.
If to summarize all that I have read and video lectures that I've watched, there are two different groups of arguments: spiritual and dogmatic. First kind of arguments is much easier to understand.
For example, here is
an interview with Jesuit Fr. Robert Taft. In which he says:
To attempt to apply rational analysis to this is to fail to understand what the East is. Once you get over on this side of the Atlantic Ocean, the further you go South or East from anywhere, the worse everything gets, except the food. Logic gets worse, rationality gets worse, and everything ultimately winds up in hysteria and emotionalism. It’s futile to try and reason about this.
To hell with Moscow.
That’s what we had before the East/West schism. Does anybody think that Rome had anything to say about who became patriarch of Constantinople? Or who became the metropolitan of Nicomedia? Of course not. These guys were bishops there just like we had bishops here, and when they met they’d say, “You’re a bishop? Hey, I’m a bishop too. How’s it going?” They were all in communion. It’s not like Rome was telling them what to do.
Do you agree that the central problem is the papacy?
Of course. What we’ve made out of the papacy is simply ridiculous. There’s no possible justification in the New Testament or anyplace else for what we’ve made out of the papacy. That doesn’t mean that I don’t believe in a Petrine ministry. I believe that Rome has inherited that Petrine ministry. But there’s no reason on God’s earth why the pope should be appointing the bishop of Peoria.
To summarize this, when people contradict to what was established by God, - they inevitably begin to contradict themselves, become evil and pride, and this ruins their souls. But the Church was established to save our souls. Another examples you can find in this article: Why Orthodoxy is the true faith
by Alexey Osipov, professor and lecturer from Moscow Orthodox Theological Seminary. On in his video lecture: "Ortodoxy and non"
. (Each video there has two links below, left link - to watch online, right - to download.) This spiritual argument he applies to both Roman Catholicism and Protestantism.