Everyone is familiar with Matthew 16:18, but consider also these verses:
"He that heareth you, heareth me; and he that despiseth you, despiseth me; and he that despiseth me, despiseth him that sent me."Luke 22:31-32:
"And the Lord said: Simon, Simon, behold Satan hath desired to have you, that he may sift you as wheat: But I have prayed for thee, that thy faith fail not: and thou, being once converted, confirm thy brethren."
"Amen, amen I say to you, he that receiveth whomsoever I send, receiveth me; and he that receiveth me, receiveth him that sent me."John 21:17:
"He said to him the third time: Simon, son of John, lovest thou me? Peter was grieved, because he had said to him the third time: Lovest thou me? And he said to him: Lord, thou knowest all things: thou knowest that I love thee. He said to him: Feed my sheep."
2 Timothy 2:2:
"And the things which thou hast heard of me by many witnesses, the same commend to faithful men, who shall be fit to teach others also."Titus 2:15-3:2:
"These things speak, and exhort and rebuke with all authority. Let no man despise thee. Admonish them to be subject to princes and powers, to obey at a word, to be ready to every good work. To speak evil of no man, not to be litigious, but gentle: shewing all mildness towards all men."
Some of these verses apply to the Apostles and/or their disciples generally. But whatever is said of the Apostles in general applies at least as much to Simon Peter, their foremost member, and however much authority the disciples of the apostles have, such as Timothy and Titus, how much more does the Prince of the Apostles!
The tradition of obeying the Successor of Saint Peter goes back to the 1st century, when Pope Saint Clement I settled an internal dispute in the Church of Corinth. The Popes regularly exercised their authority in the early church, and opposition was scant.
Obedience to the Pope was repeatedly affirmed in the official teachings of the Church. The Formula of Hormisdas. Unam Sanctam. The Fifth Lateran Council. Vatican I. Vatican II.
The saints have repeatedly affirmed this as well:
Saint Thomas Aquinas:
"The administration of this kingdom, therefore, is entrusted not to earthly kings, but to priests, so that spiritual and earthly things may be kept distinct; and in particular to the Supreme Priest, the successor of Peter, the Vicar of Christ, the Roman Pontiff, to whom all the kings of the Christian people should be subject, as if to the Lord Jesus Christ Himself."Saint Catherine of Siena:
"For divine obedience never prevents us from obedience to the Holy Father: nay, the more perfect the one, the more perfect is the other. And we ought always to be subject to his commands and obedient unto death. However indiscreet obedience to him might seem, and however it should deprive us of mental peace and consolation, we ought to obey; and I consider that to do the opposite is a great imperfection, and deceit of the devil."Saint John Bosco:
"Do not ever forget these three things: devotion to the Blessed Sacrament, devotion to Mary Help of Christians, and devotion to the Holy Father!"Blessed Ivan Merz:
"Why do I love the Catholic Church and the Pope? Because in the Church I can see a clear image of the beloved Saviour, the God-man, Jesus Christ, with all his perfections, and in the figure of the Pope I can see my God and my Lord."Nothing could be more clear from the tradition of the Church: Obey the Pope!
Even if you aren't Christian, you can't help but see in the Pope the betterment of humanity. Rome was sacked by barbarians in the 5th century. The empire fell, but the Pope remained, and the barbarians became civilized under the teaching and guidance of the Pope. Far from being a religious conquest, this was the largest, longest, most widespread peaceful conversion in human history. During the Middle Ages, the Pope was frequently the only check on the power of Europe's monarchs. Moreover, the medieval wars when Europe was united under the Pope paled in comparison to the horrors of the wars that broke out after Europe abandoned the Pope in the 16th, 17th and subsequent centuries. The Pope resisted Napoleon and was a voice for peace in WW1 and WW2. 20th and 21st century Popes have consistently stood out as voices against racism, nuclear weapons and the death penalty.
Last but not least, obedience to the Pope instills virtue and banishes vice. Saint Thomas Aquinas wrote that obedience is the greatest of the moral virtues:
Among the moral virtues, the greater the thing which a man contemns that he may adhere to God, the greater the virtue. Now there are three kinds of human goods that man may contemn for God's sake. The lowest of these are external goods, the goods of the body take the middle place, and the highest are the goods of the soul; and among these the chief, in a way, is the will, in so far as, by his will, man makes use of all other goods. Therefore, properly speaking, the virtue of obedience, whereby we contemn our own will for God's sake, is more praiseworthy than the other moral virtues, which contemn other goods for the sake of God.Think of the theological virtues and their opposites, and whether obedience to the Pope helps or hinders them:
Faith: Obedience to the Pope is first and foremost an act of faith. It is an act of trust that the Pope truly is the Vicar of Jesus Christ and that the Holy Spirit is guiding him. Disobeying the Pope, on the other hand, is an act of unbelief in the promise of Jesus Christ to Saint Peter. Saint Thomas Aquinas calls unbelief the greatest of the theological vices.
Hope. When we obey the Pope, it is with great hope that God will bless both the Pope and our actions in obedience to his commands. Knowing that Jesus Christ is building his Church on the Pope, how great must our expectation be of the fruit that will come from obeying His Vicar! Disobedience of the Pope, on the other hand, tends to the vice of presumption, either in the sense that we think that we have greater wisdom than God's chosen leader, or in the sense that we overlook the significance of disobeying Christ's representative on Earth. This can only lead to the polar opposite of hope - despair. For if God has not kept his promise to guide the Successor of Saint Peter with sound wisdom and leadership, then what hope is there for anyone else?
Charity. When we obey the Pope, we are set free from useless quarrels, schism and heresy. We have full confidence that we are doing the Lord's work and can set about it with complete confidence, knowing that God will bless our actions. Saint Thomas lists the vices opposed to charity: hatred, sloth, envy, discord, contention, schism, war, strife, sedition and scandal. It is all too obvious how disobedience to the Vicar of Jesus Christ leads to discord, contention, schism and strife.
Humility. Saint Thomas Aquinas wrote, "Therefore after the theological virtues, after the intellectual virtues which regard the reason itself, and after justice, especially legal justice, humility stands before all others." When we obey the Pope, it is an act of humility. We are acknowledging that someone else knows better than us. More than this, we are acknowledging that God has chosen someone other than us to be in charge, and we are ok with that. Disobedience to the Pope is fundamentally an act of pride - the most grievous of all sins. Obedience is the antidote to pride. To disobey the Pope is to dare to say that we know better than the Vicar of Jesus Christ. It is the first sin, the sin of Adam and the sin of Lucifer, to think so highly of ourselves that we are not merely disobeying God's representative, but in doing so, disobeying God Himself.