Not even remotely. According to patristics scholar J. N. D. Kelly, of the period that spans the church fathers, from the first through 7th centuries, only roughly 1/3rd of the fathers understood the passage to be referring to Peter. The other 2/3rds interpreted it either as his confession of faith, or the rock being Christ himself. Since no one in the early church period (especially the anti-Nicene period) ever conceived of the modern notions of a papacy, even those who do interpret the passage as referring to Peter, never connect it with later papal claims. The best thing to do, though, is go to the horses mouths, as it were. First, examine an early church theologian who did interpret the passage as referring to Peter, Tertullian:
“Was anything withheld from the knowledge of Peter, who is called the ‘rock on which the church should be built’ who also obtained ‘the keys of the kingdom of heaven,’ with the power of ‘loosing and binding in heaven and earth? (Alexander Roberts and James Donaldson, Ante-Nicene Fathers (Grand Rapids: Eerdmans, 1951), Volume III, Tertullian, Prescription Against Heretics 22).”
“If, because the Lord has said to Peter, ‘Upon this rock I will build My Church,’ ‘to thee have I given the keys of the heavenly kingdom;’ or, ‘Whatsoever thou shalt have bound or loosed in earth, shall be bound or loosed in the heavens,’ you therefore presume that the power of binding and loosing has derived to you, that is, to every Church akin to Peter, what sort of man are you, subverting and wholly changing the manifest intention of the Lord, conferring (as that intention did) this (gift) personally upon Peter? ‘On thee,’ He says, ‘will I build My church;’ and, ‘I will give thee the keys’…and, ‘Whatsoever thou shalt have loosed or bound’…In (Peter) himself the Church was reared; that is, through (Peter) himself; (Peter) himself essayed the key; you see what key: ‘Men of Israel, let what I say sink into your ears: Jesus the Nazarene, a man destined by God for you,’ and so forth. (Peter) himself, therefore, was the first to unbar, in Christ’s baptism, the entrance to the heavenly kingdom, in which kingdom are ‘loosed’ the sins that were beforetime ‘bound;’ and those which have not been ‘loosed’ are ‘bound,’ in accordance with true salvation…(Alexander Roberts and James Donaldson, The Ante-Nicene Fathers (Grand Rapids: Eerdmans, 1951), Volume IV, Tertullian, On Modesty 21, p. 99).”
When Tertullian says that Peter is the rock and the Church is built upon him he means that the Church is built through him as he preaches the gospel. This preaching is how Tertullian explains the meaning of the keys. No mention of a papacy, primacy, infallibility, etc. Of the fathers that disagreed with Peter being the rock. Another father who interprets it as being Peter, Cyprian:
“The Lord saith unto Peter, I say unto thee, (saith He,) that thou art Peter, and upon this rock I will build My Church, and the gates of Hell shall not prevail against it. And I will give unto thee the keys of the kingdom of heaven, and whatsoever thou shalt bind on earth, shall be bound in heaven, and whatsoever thou shalt loose on earth, shall be loosed in heaven (Matt. 16:18–19). To him again, after His resurrection, He says, Feed My sheep. Upon him being one He builds His Church; and although He gives to all the Apostles an equal power, and says, As My Father sent Me, even so I send you; receive ye the Holy Ghost: whosoever sins ye remit, they shall be remitted to him, and whosoever sins ye shall retain, they shall be retained (John 20:21);—yet in order to manifest unity, He has by His own authority so placed the source of the same unity, as to begin from one (A Library of the Fathers of the Holy Catholic Church (Oxford: Parker, 1844), Cyprian, On The Unity of the Church 3-4, pp. 133-135).”
Later he says: “Our Lord whose precepts and warnings we ought to observe, determining the honour of a Bishop and the ordering of His own Church, speaks in the Gospel and says to Peter, I say unto thee, that thou art Peter, and on this rock I will build My Church; and the gates of hell shall not prevail against it. And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven. Thence the ordination of Bishops, and the ordering of the Church, runs down along the course of time and line of succession, so that the Church is settled upon her Bishops; and every act of the Church is regulated by these same Prelates (A Library of the Fathers of the Holy Catholic Church (Oxford: Parker, 1844), The Epistles of S. Cyprian, Ep. 33.1).”
continued.