I've written on this topic before which I'll repost below:
The idea that Columbus was a lone genius stubbornly striving for truth against a world insisting he was wrong about the shape of the globe is an idea which stems from Enlightenment era conceptions about the so called "Dark Ages", a vision of the medieval period in which the knowledge of the ancient Greeks was lost to mankind, and instead human progress was severely curtailed by the superstitious and backwards Catholic Church. This is bunk on a number of levels. Our FAQ has several entries which dismiss the antiquated view of the middle ages as "the Dark Ages", and I myself have touched on the fact that Columbus wasn't thought a fool because of his belief in a round earth, but rather because he severely underestimated the circumference of the globe. In brief, the reality is that he was the one who was wrong, and everyone else was right, and he only 'succeeded' by sheer dumb luck that there happened to be an unknown continent for him to run into. People thought he was an idiot because they thought he was going to have to sail to Asia through the Atlantic and the Pacific as if the Americas didn't exist. That is a long journey and he never would have made it.
Anyways though, that is digressing a little. To get back on track, as I said, this idea of Columbus being the lone beacon of truth is grounded in the Enlightenment view of the middle ages, and while it doesn't seem entirely clear that he was the absolute first person to propose it, it is generally agreed that Washington Irving's 1828 "The Life and Voyages of Christopher Columbus" was the book that pushed the idea into the popular mindset. Irving's biography was a smash hit, and he was given numerous awards in recognition of the work, most notably in the circumstances membership in the Real Academia de la Historia.
The work itself is remembered for painting a stirring - and completely fictional - picture of Columbus pleading his case to Ferdinand and Isabella in the face of obstinate resistance from the clergy, although in truth, Irving didn't make quite the absolute image that was remembered from it, as the passage in full reads:
The passage cited from Lactantius to confute Columbus is in a strain of gross ridicule, unworthy of so grave a theologian. " Is there anyone so foolish," he asks, "as to believe that there are antipodes with their feet opposite to ours ; people who walk with their heels upward and their heads hanging down that there is a part of the world in which all things are topsy-turvy ; where the trees grow with their branches downward, and where it rains, hails, and snows upward? The idea of the roundness of the earth," he adds, "was the cause of inventing this fable of the antipodes with their heels in the air ; for these philosophers, having once erred, go on in their absurdities, defending one with another." More grave objections were advanced on the authority of St. Augustine. He pronounces the doctrine of antipodes incompatible with the historical foundations of our faith ; since, to assert that there were inhabited lands on the opposite side of the globe, would be to maintain that there were nations not descended from Adam, it being impossible for them to have passed the intervening ocean. This would be, therefore, to discredit the bible, which expressly declares, that all men are descended from one common parent.
As implied by the passage, Irving does present some objectors lampooning the idea of a round earth, but others objecting simply to the idea that there could be inhabited lands on the other side of it (a somewhat more realistic objection, touching on a debate that was quite real if you see the linked answer). Irving even goes further to note that:
Others, more versed in science, admitted the globular form of the earth, and the possibility of an opposite and inhabitable hemisphere ; but they brought up the chimera of the ancients, and maintained that it would be impossible to arrive there, in consequence of the insupportable heat of the torrid zone. Even granting this could be passed, they observed, that the circumference of the earth must be so great as to require at least three years to the voyage, and those who should undertake it must perish of hunger and thirst, from the impossibility of carrying provisions for so long a period. He was told, on the authority of Epicurus*, that, admitting the earth to be spherical, it was only inhabitable in the northern hemisphere, and in that section only was canopied by the heavens ; that the opposite half was a chaos, a gulph, or a mere waste of water. Not the least absurd objection advanced, was, that should a ship even succeed in reaching, in this way, the extremity of India, she could never get back again ; for the rotundity of the globe would present a kind of mountain, up which it would be impossible for her to sail with the most favourable winds.
But nevertheless, there remain several issues here. In the first, while it is certainly clear that Irving didn't present universal condemnation of the round earth, he certainly illustrated it as only the learned men of science who unequivocally agreed, and further he presents many of the objections as still being quite absurd. More importantly though, this wasn't the part that stuck in peoples' memory. That, of course, was the bit about a flat earth, and it only amplified over time. Several works that came out soon after cited Irving's claim helping to give further credence, and this was especially helped by the article by the anti-clerical scholar Antoine-Jean Letronne entitled "On the Cosmological Opinions of the Church Fathers" which heavily pushed the idea that flat-eartherism was near universal within Church beliefs through the Middle Ages, and thus gave more force to the idea. To be sure, there were a few ancient theologians who could be cited in support, including the 3rd century writer Firmianus Lactantius and the 6th century Cosmas Indicopleustes, but they were a distinct minority, and not at all representative of the body of thought, however these 19th century authors might have painted them. By 1853, hedging such as we see in Irving's account was mostly gone, as we can see in this passage from Alphonse de Lamartine "Life of Columbus" where he clearly draws on Irving but nevertheless writes rather unequivocally:
To this council the King had added the professors of astronomy, of geography, of mathematics, and of all the sciences taught at Salamanca. The audience did not alarm Columbus. He expected to be tried by his peers, but he was only tried by his despisers. The first time he appeared in the great hall of the convent, the monks and so-called wise men, convinced beforehand that all theories surpassing their ignorance or their routine were but the dreams of a diseased or arrogant mind, saw in this obscure foreigner only an adventurer seeking his fortune by these chimeras. None deigned to listen to him, save two or three friars of the convent of St. Stephen of Salamanca, obscure monks without any influence, who devoted themselves in their cells to studies despised by the superior clergy. The other examiners of Columbus puzzled him by quotations from the Bible, the prophets, the psalms, the Gospels, and the fathers of the Church ; who demolished by anticipation, and by indisputable texts, the theory of the globe, and the absurd and impious idea of antipodes. Amongst others, Lactantius had expressed himself deliberately on this subject in a passage which was cited to Columbus: "Can anything be more absurd,'' Lactantius writes, "than to believe in the existence of antipodes having their feet opposed to ours — men who walk with their feet in the air and their heads down, in a part of the world where everything is topsy-turvy — the trees growing with their roots in the air, and their branches in the earth?" St. Augustine had gone further, branding with impiety the mere belief in antipodes : ** For," he said, ''it would involve the supposition of nations not descended from Adam. Now, the Bible says, that all men are descended from one and the same father." Other doctors, taking a poetical metaphor for a system of cosmogony, quoted to the geographer the verse of the psalm in which it is said that God spread the sky above the earth as a tent— from which it followed, they said, that the earth was flat.
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As it happens with many anti-Christian (or rather anti-Catholic) myths or black legend, it comes from the early 19th century, the moment when the Catholic Church started to lose power in a substantial way. That is the time of the French Revolution, of the Enlightment, of early modern science. We can't pinpoint the myth to one author in particular, but there are two names that rise above everyone else in the propagation of this myth: the extremely popular author Washington Irving, and French archaelogist Letronne.
In Washington Irving's very novelistic biography of Christopher Columbus, which is full of myths, misconceptions, and inconsistencies, we see the idea that Columbus sailed to prove the Earth was round, unlike what the Church affirmed which is the Earth being flat. Washington Irving was, after all, a novelist, not a historian; he was no Marquess of Fuensanta del Valle or Martín Fernández de Navarrete. Through Irving's popularity, the myth became "common knowledge". The worse part is that Irving contradicts himself when he affirms that the question was not about the Earth's shape but about its size and the distance to the Indies.
Antoine-Jean Letronne, on the other hand, was a respected archaeologist, but he had a heavy anti-Catholic bias. Letronne wrote in 1834 an article in the very reknowned and respected "Revue des Deux Mondes" where he claimed the Catholic Church believed the Earth was flat throughout the Middle Ages, and to that of course he opposes the ideals of modernity.
From the very popular Washington Irving, and with the support of Letronne's article against the Church, the whole idea started rolling, and from then on we have had this myth in circulation.
As we await an in-depth answer, you might find /u/Georgy_K_Zhukov's answer in a different thread about Columbus interesting. He briefly mentions the Catholic Church in regards to flat-earthism in the below thread.