This is a tricky question, because it is nearly as much a linguistic and theological one as a historical one. Much of the earliest history necessarily comes from the Gospels, which are usually not the right sources to use in /r/askhistorians. I will attempt to confine myself to the historical matter.

In around 30 AD, Jesus the Nazarene, who (according to reports whose veracity is beyond the scope of this question) had recently returned from the dead, radically altered the scope of his ministry.

Prior to his crucifixion, all accounts suggest that Jesus had been a teacher and reformer within the Jewish rabbinic tradition. He drew followers to him, he preached his revolutionary interpretation of the Hebrew Scriptures -- revolutionary both in the sense that he openly embraced eternal life (a controversial proposition among contemporary Jews) while downplaying the letter of the Mosaic Law, and in the sense that he directly attacked the powerful Pharisees as snakes and hypocrites -- and became a notable figure, but Jesus did not evangelize. His Apostles did not convert anyone. In fact, mostly they sat around taking private lessons and making asses of themselves. In fact, on more than one occasion, Jesus specifically forbade his followers from spreading what would come to be known as the "Good News." The only hint we see of the coming Apostolic Age was Jesus's promise to Simon Peter in Matthew 16:

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 to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose upon earth, it shall be loosed also in heaven.

After the Resurrection, however, Jesus suddenly becomes quite the missionary. This is when we get "Go and make disciples of all nations". This is when the Holy Spirit descends (allegedly), and when the Apostles start running around becoming martyrs and preaching Jesus. This is when the personal cultus of Jesus becomes the Christian Church.

This Christian Church appears to have been led, to a limited extent, by Peter. Peter had been given the keys to the Kingdom, mentioned above. In one of the rare instances of disagreement among the apostles and elders (over circumcision), it is Peter who rises and ends the dispute, citing his own Christ-given authority, and the other defer to him (Acts 15). When Judas needs to be replaced among the order of Apostles, it is Peter who presides over the meeting (Acts 1).

1 Peter 5:13 seems to confirm that Peter was heading up the Church in Rome, and this is attested to (hooray for extra-biblical attestation, finally!) by Ireneaus, Tertullian, and Clement during the second century (Clement writing in 96 AD, just a few decades after Peter's death). Moreover, we now have Peter's tomb, uncovered beneath the Vatican. I think there's a strong case to be made that the tomb really is St. Peter's (Margherita Guarducci makes this case here; please forgive the ugly site and focus instead on the author's credentials), but what is unambiguously established by the scavi is that, by the third century (still decades before Constantine will even be born), Christians were making pilgrimages to the Roman tomb of St. Peter at Vatican Hill.

When Peter died, Irenaeus (writing around 180 AD) records that the saint committed his authority to St. Linus, a martyr whose service as Peter's successor makes him (in Catholic records) the second pope. The pre-eminence of the Roman see continued to be recognized by the Church throughout the Patristic Age. Ignatius of Antioch refers to the Roman Church as "president" of the universal Church. The Shepard of Hermas (whatever its other merits) recognized that Rome was chiefly responsible for "disseminating" the Good News. Ireneaus unambiguously condemns all those who disagree with the Church of Rome as heretics (III.3.2). Even when the Bishop of Rome did stupid things (like a mass excommunication Eusebius records here, in Chapter 24), his authority to do stupid things was recognized. And, of course, centuries later, St. Augustine will sum up the era in his famous statement that, once "the Apostolic See" has spoken consistently, with authority, on a matter of faith, "the cause is finished" (often paraphrased loosely as "Roma locuta est; causa finita est").

To be sure, the organization then known as the Christian Church would evolve and further define itself. In the early centuries, the Roman authority was not seen as an absolute sovereign (nor did Peter use it as such), but as "first among equals", only empowered to exercise authority over other sees after careful consideration when vital to protecting the Faith -- as when Pope Stephen overruled the African bishops who required rebaptism, and not so much when Zosimus rashly proclaimed Pelagius orthodox (without really knowing Pelagius's actual teachings). The Church's doctrines were shaped primarily by ecumenical councils (starting with the Council of Jerusalem in Acts), with popes more rarely attempting to act on their own.

Nevertheless, it is pretty clear that the organization Christ founded with Peter has unambiguously survived into today. Peter and his fellow Apostles have been succeeded by bishops, selected according to processes established and sanctioned by those same bishops in an unbroken succession. The bishopric of Rome has retained its pre-eminence within the organization. The authority of ecumenical councils, convoked by those bishops, is still unquestioned within that organization. And there is no one else, other than Jesus, who can be plausibly named as the founder of the organization.

That organization is the Catholic Church.

Note that this does not only include the Roman Catholic Church. The Eastern Orthodox, while estranged from Rome, can mostly say the same things: they are founded on the apostolic succession, they respect Rome at least as "first among equals," they recognize the councils (but disagree about what councils were validly convoked), and it's impossible to point to somebody other than Jesus who founded the Eastern Orthodox. The Great Schism in the Catholic Church was like cell division: where at first there was one organization, eventually there came to be two organizations. Which is the "real" one? To some extent, both are real, as both organizations recognize insofar as they accept the validity of each others' sacraments.

But very few schismatic groups can make that claim. Most Christian groups that broke off from the Catholic Church did so by clearly and discretely abandoning some key element of the governing structure which the Apostles had established in Jesus's name, and can point to a single man (or small group) who led the breakaway, like Martin Luther for the Lutherans, or Marcel Lefebvre for the Lefebvrists. Many in the Protestant Reformation abandoned the entire governing structure -- the apostolic succession, Roman primacy, holy orders itself, and any definitive teaching authority beyond personal interpretation of the Scriptures -- all at the same time!

In short, yes, Jesus personally established the Catholic Church, but this does not include solely the Latin Church with which we are all most familiar. It clearly includes both the Eastern Orthodox and the Eastern Catholic Churches, and arguably could include even certain high-church elements of the Anglican Communion, who have struggled to maintain the apostolic succession (and arguably succeeded, although Rome does not agree).

This is not to say that Jesus would have approved or sanctioned everything that the Church he founded has done in the centuries since then. Plenty of old organizations have changed a lot as they've been passed down over decades, and it's hard to imagine, for instance, that William Lloyd Garrison would be a big fan of what the magazine he sorta founded (The Nation) has become today. Maybe Jesus would have hated the idea of becoming the state religion of Rome (which, incidentally, contrary to popular belief, happened in 380, fifty years after the death of Constantine -- who legalized Christianity but did not mandate it). Maybe Jesus would have supported contraception for married couples. Maybe Jesus would agreed that Eucharist is purely symbolic and laughed at the Real Presence doctrine of Catholics. The fact that Catholicism was founded by Jesus does not necessarily mean Catholicism is correct about Jesus. But these huge doctrinal questions fall way outside the scope of your question -- and, indeed, of this subreddit.