From my understanding, the Catholic belief of ordained church leadership comes from the interpretation of Jesus' words to Peter:
And I tell you, you are Peter, and on this rock I will build my church, and the gates of hell shall not prevail against it. - Matthew 16:18 ESV
15 When they had finished breakfast, Jesus said to Simon Peter, "Simon, son of John, do you love me more than these?" He said to him, "Yes, Lord; you know that I love you." He said to him, "Feed my lambs." 16 He said to him a second time, "Simon, son of John, do you love me?" He said to him, "Yes, Lord; you know that I love you." He said to him, "Tend my sheep." 17 He said to him the third time, "Simon, son of John, do you love me?" Peter was grieved because he said to him the third time, "Do you love me?" and he said to him, "Lord, you know everything; you know that I love you." Jesus said to him, "Feed my sheep. - John 21:15-17 ESV
In light of that, how does the Catholic Church interpret Paul's division of gospel responsibility as summarized in Galatians 2:7, and as carried out in the other letters to the churches, as the Roman Catholic Church is a gentile church?
6 And from those who seemed to be influential (what they were makes no difference to me; God shows no partiality)-those, I say, who seemed influential added nothing to me. 7 On the contrary, when they saw that I had been entrusted with the gospel to the uncircumcised, just as Peter had been entrusted with the gospel to the circumcised 8 (for he who worked through Peter for his apostolic ministry to the circumcised worked also through me for mine to the Gentiles), 9 and when James and Cephas and John, who seemed to be pillars, perceived the grace that was given to me, they gave the right hand of fellowship to Barnabas and me, that we should go to the Gentiles and they to the circumcised. 10 Only, they asked us to remember the poor, the very thing I was eager to do. - Galatians 2:6-10 ESV