The answer is yes. God did intend for there to be one united Church led by leadership that Christ set forward before and after His death and resurrection. Let me show you some Scriptural proof for these statements, which are very much in favor of the Catholic position.
In Matthew 16:13-19 we see that Jesus renames Simon to Peter and then tells him that he is head of the Church, gives him the keys to the kingdom of heaven and tells him whatever he binds on earth shall be bound in heaven and whatever he looses her on earth shall be loosed in heaven, and also proclaims that the gates of hell will not prevail against the Church. Later Peter moves to Rome and he becomes the bishop of Rome and headquarters the Church in Rome. Anyone who does not accept Peter or his successors authority (Apostolic Succession is the next paragraph) must obviously not be following the Church that Jesus founded.
The previous paragraph mentions Peter’s successors. How do we know that the power that the 11 Apostles were given could be passed down to other people that were not named as Jesus’ disciples? We look to Acts for that answer. Acts 1:15-26 describes the first Apostolic Succession when Matthias was chosen to take Judas’ place. Verse 26 is especially important. It says, “and he was counted with the 11 Apostles.” That verse states that he was counted with the other Apostles. In other words Matthias was equal to the rest of the Apostles.
He had the same abilities and the same powers that Jesus bestowed upon the original Apostles even though he was never one of the original Apostles. There is only one Church that has maintained a continuous Apostolic Succession with the Bishop of Rome as its head and that is the Catholic Church. Step 2: if you do not follow a church with an Apostolic Succession you are not following the Church that Jesus founded.
How do we know that the Apostles were making the right decision when they chose successors for themselves? We return to Matthew and read about the commissioning of the disciples. This is in Matthew 28:16-20. In verses 18 and 19 Jesus proclaims that, “All power in heaven and on earth has been given to me. Go, therefore, and make disciples of all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Spirit.” Jesus first declares He has an enormous amount of power. Then he sends forth the Apostles to do some work for Him. He says, “Go, therefore.” Since all universal power belongs to Jesus he gives them a mission that is universal. They are to make disciples of all nations. “All nations” means everyone. It not only means Gentiles, but Jews also. Remember that God is the god of all people Gentile or Jew (and He, of course, is still the God of all people no matter what beliefs you have).
We skip to the very important part of this passage that helps to make the point for this essay. “Teaching them to observe all that I have commanded you. And behold, I am with you always, until the end of the age,” is Matthew 28:20. The second part of that sentence is key to the point because Jesus promises that He will be with them always until the end of the age. Since Jesus promised this, then the decisions that they made in regards to the direction of the Church must be the right direction that Jesus wanted the Church to go. So apparently Jesus wanted successors to the Apostles and to spread His message all over the world. Besides, why would Jesus set something up that would die out after the first generation? The age has not ended for Jesus has not yet come again.
Adding on the evidence listed above for the question “How do we know that the Apostles were making the right decision when they chose successors for themselves?” is in Matthew 16:18. In that verse a couple of things happen as they have already been named. After Jesus renames Simon to Peter (rock) and tells him that he is the leader of the Church he then proceeds to say that, “the gates of hell will not prevail against it [the Church].” Jesus, along with promising that He will always be with the Apostles, tells Peter that evil and corruption will not prevail against the Church that Peter leads. This means that the Church will be free from teaching error as it cannot be corrupted in what it teaches because, well, Jesus said so.
In Matthew 16:19 Jesus gives Peter the “power of the keys” after he proclaims him the leader of His Church. Two chapters later, Matthew 18:18 Jesus gives authoritative power to all of the Apostles. You will notice that this verse parallels Matthew 16:19, however in Matthew 18:18, the power bestowed on the Apostles is a bit more limited than Peter’s. Authority is granted to all of the Apostles, but none of them except for Peter are declared leader of the Church in this verse.
To better understand “the power of the keys” we look to the Old Testament for this reference. In Isaiah 22:15-25 we have the first example of the keys being bestowed upon someone else and having them represent a transfer of power. In this passage Eliakim, who succeeds Shebnah as master of the palace, is given “the key to the house of David,” which he authoritatively “opens” and “shuts.” The key verse is Isaiah 22:22, but the previous verses must also be taken into account as with anything in the Bible. That verse says, “I will place the key of the House of David on his shoulder; when he opens, no one shall shut, when he shuts, no one shall open.” In other words, Eliakim has the final say in what goes on in his kingdom. This is what is meant in Jesus’ time with the “power of the keys.” If you recall, Jesus was a Jew and so were all of the disciples until Christ came. The only Scripture they had at the time were the books of the Old Testament. There are many times where Jesus uses references to these books and this is one of them. When the “power of the keys” is transferred to someone it is a big deal and that person then has an extremely high place of authority.