These are the arguments a Catholic will meet in practice, stated as they are made, with the answer and the ground it stands on. Two habits make the conversation worthwhile: put the claim in its strongest form before answering it, and keep the argument in the texts and the historical record rather than in personalities.
The claim
“On this rock” means Peter's confession of faith, not Peter. Augustine says so.
The answer
Augustine says both at different times, and in the same years he describes the see of Peter as the rock and appeals to it. But the decisive point is not Augustine; it is the language Christ spoke. In Aramaic there is one word, kepha, so He said: you are Kepha, and on this kepha. And the grant that follows in the very next verse is personal — the keys, and the power to bind and loose. It would be strange for Christ to build His Church on a confession and then hand the government of it to the man who made it.
Matthew 16:17-19 · John 1:42
St Jerome, Letter 15.2: the Church is built on that rock
The claim
The power to bind and loose was given to all the apostles (Matthew 18:18; John 20:23), so Peter had no primacy.
The answer
Both are true and they are ordered. In the household of David the steward held the keys and the elders shared in the governance of the house. Matthew 16 gives the keys of the kingdom to one man; Matthew 18 gives the household its share of authority. Scripture does not set these against each other, and the apostolic practice of Acts 15 shows the structure working: the college debates, Peter pronounces, silence follows, and James then settles the practical terms.
Matthew 16:19; 18:18 · Acts 15:6-11
The claim
Constantine invented the papacy in the fourth century.
The answer
The Church of Rome wrote to Corinth and instructed it how to settle its dispute around AD 96, when the emperor Constantine was not yet born. Ignatius, Irenaeus, Tertullian, Cyprian and Optatus describe the Roman see's authority in the second and third centuries. Constantine convened the Council of Nicaea; he did not create the see of Peter, and he died a catechumen — the bishop he was to have been baptised by was an Arian.
1 Clement 42-44 (c. AD 96)
Irenaeus III.3.2 · Cyprian, On the Unity 4
The claim
Paul rebuked Peter to his face at Antioch, so Peter was not the chief.
The answer
Paul rebuked Peter for a failure of conduct, not of teaching, and the rebuke was about the truth of the gospel in practice — which is why Paul could write that he had laid his gospel before Peter and the other leaders (Galatians 2:2) and could speak of Peter's apostolate to the circumcised as equal to his own to the Gentiles (Galatians 2:7-8). The Church has never claimed that popes are sinless or beyond correction. What Christ promised Peter was that his faith would not fail, not that his behaviour would always be consistent.
Galatians 2:2, 7-11 · Luke 22:32
The claim
Nobody called the bishop of Rome “pope” in the early Church.
The answer
Vocabulary is not the office. The title 'papa' comes into Christian usage in the third and fourth centuries, and Tertullian was already mocking the Roman bishop as pontifex maximus around AD 220. What matters is that the function is documented under other names from the first century onward — Rome's intervention in Corinth, Irenaeus' pre-eminent authority, Cyprian's chair of Peter, Optatus' episcopal chair first conferred on Peter in Rome. Titles arrive late because they arrive when controversy makes precision necessary.
2 Thessalonians 2:15
Optatus, Against the Donatists II.2 (c. AD 370)
The claim
Vatican I created the doctrine of infallibility in 1870.
The answer
The practice is fifteen hundred years older than the definition. In AD 451 the bishops of Chalcedon acclaimed the letter of Pope Leo with the cry that Peter had spoken through Leo; Augustine said of a Roman judgement that the case was closed, in a sermon preached about AD 400; the reunion formula of AD 519 confessed that the Catholic religion had always been kept undefiled in the Apostolic See. What 1870 did was define the conditions of a guarantee the Church already trusted.
Matthew 16:18-19 · 1 Timothy 3:15
Council of Chalcedon, AD 451
The claim
The Donation of Constantine was a forgery, so Rome's claims rest on forgeries.
The answer
The Donation was indeed exposed as a forgery, and it was exposed by Catholic scholars: Lorenzo Valla in the fifteenth century, with Nicholas of Cusa pressing the case. The episode is a credit to the Church's willingness to be corrected by evidence, not a stain on the doctrine — because the Donation plays no part in the argument for the primacy, which rests on Scripture and on the first-century record. A forged document pretending to give the papacy land cannot create or destroy an office attested in AD 96.
The claim
Honorius I was a heretic pope, so popes can teach error.
The answer
This objection was argued at Vatican I itself, and the answer given then still holds. Honorius wrote private letters to Sergius in which he failed to condemn the heresy of monotheletism; he was condemned by a council for negligence, not for a formal definition. He taught nothing ex cathedra to the whole Church, which is the only case the charism covers. The episode actually shows the narrowness of the claim: if infallibility meant what its critics say it means, Honorius would destroy it.
Matthew 16:18
Vatican I, Pastor Aeternus, ch. 4
The claim
The Renaissance popes were corrupt, so this cannot be the Church of Christ.
The answer
The Lord Himself answered this in advance. He told His disciples to obey those who sat on Moses' seat, while warning them not to imitate what those men did (Matthew 23:2-3). The authority belonged to the seat, not the sitter. The same objection, applied consistently, would falsify the apostles themselves: one betrayed Him, one denied Him three times, and all but one fled. Christ promised indefectibility in faith, not personal holiness in office.
The claim
The East never accepted the papacy; Rome imposed it.
The answer
The East appealed to Rome repeatedly — Athanasius in the fourth century, John Chrysostom when he was deposed in 404, Flavian of Constantinople when he was condemned in 449. The entirely Eastern Council of Constantinople in 381 gave its own bishop the primacy of honour after the bishop of Rome, a canon that presupposes Rome stands first. What the East resisted in the eleventh century was a particular claim about the manner of jurisdiction, not the fact that Rome held the first see.
Acts 15:7-11
Council of Constantinople I, canon 3 (AD 381)
The question underneath. Strip away the history and one question remains: did Christ leave His Church a visible centre of unity, or did He leave it a book and a disagreement? He prayed that they might be one so that the world would believe, He renamed a man Rock, gave him the keys of the house of David, prayed that his faith would not fail, and told him to feed His sheep. The Catholic claim is that He meant it.