I'm currently reading "Columbus - The Four Voyages", by Laurence Bergreen. I don't have a clear answer for you, but some of the things that I've read might be of interest to you.
Columbus was excited by the prospect of facing the people that didn't believe in him. In his diary he talks about "the opinion of so many high personages... who were all against me, alleging this undertaking to be folly".
One of his main detractors, King Joao II from Portugal, had to deal with the new papal decree signed by Pope Alexander VI. The pope had followed closely the news about Columbus' discoveries and was interested in increasing the reach of Catholicism (and his own personal power). He then divided the lands between Spain and Portugal, infamously "drawing a line 'one hundred leagues toward the west and south from any of the islands commonly known as the Azores and Cape Verdes.' Anything west of that line--that is, everything that mattered--was Spain's, and if Columbus was able to complete another voyage, partly his as well."
"While Columbus prospered, King Joao II of Portugal feverishly tried to reverse the papal decree, which threatened to diminish the Lusitanian empire. After enduring threats of naval action and pleas for Iberian cooperation for a year, Joao II cajoled Ferdinand and Isabela to send representatives to a summit conference in Tordesillas, Spain, on June 7, 1494, where a treaty between the two sovereign power shifted the line of demarcation", famously providing the Portuguese crown with a claim over Brazil.
EDIT: Spelling mistake. Thanks to u/pinguis.
On the whole, Spain was ecstatic and found ways to justify the weird geography of the voyages. Portugal, which had turned Columbus away entirely, was devastated.
Columbus was a national sensation when he returned to Spain, to say the least. He did a good job of getting the word out, actually going so far as to drop a barrel with a record of his voyages in the Atlantic, in case something happened to him before he could report it in Spain. Even when he arrived in Lisbon (his first continental port, and enemy territory!), he sent out letters to Spain and didn't waste time telling King João II what Portugal had missed out on. Once Columbus arrived in Palos, Spain, he drafted a full report of his voyages and delivered it to the royal court.
The letter only survives in a copy that was carefully edited by the royal court, mainly in order to avoid any suggestion that the new lands might belong to Portugal. But even with the edits, it's clear that Columbus was heavily exaggerating his findings. He described the people as gentle and generous; the islands were full of incalculable gold just waiting to be mined; and the fabulously wealthy cities of the Far East were just beyond the islands. Columbus actually suggested that the islands might be somewhere near the Terrestrial Paradise, the preserved Garden of Eden that medieval Christians believed was in the East. This letter was immensely popular, especially outside royal circles. The first edition letter was published in Castillian in April 1493, then translated into Latin. By the end of 1493, there had been three new editions from Barcelona, three from Paris, and others in Antwerp and Basel. There was even an Italian rhyming version published in Rome.
While no one in Spain could personally verify Columbus' claims, he brought back some important New World specimens: natives, parrots, spices, etc. This is what really got people excited - wherever Columbus and his retinue traveled, people would flood the streets to see the exotic arrivals. For those who doubted Columbus' plan, the material benefits were impossible to deny when they were being physically paraded across Spain.
But this of course doesn't directly address the reactions of the individuals who had doubted Columbus. First, the Portuguese: the crown had rejected Columbus, and the story goes that King João II was devastated by the news of his success. There's an apocryphal account (probably invented by Bartolome las Casas) that reports the king's outburst (Historia de las Indias 1.74):
Then the king, understanding clearly the greatness of the discovered lands, and the riches that had already been imagined in them, not being able to hide the great pain which he felt...for the loss of such inestimable things, which through his own fault he had let slip through his hands, with a loud voice and an impulse of anger against himself, beat his fist on his chest saying: "Oh, man of poor understanding, why did you allow such an enterprise of such importance to get out of your hands?"
Again, this is coming from Las Casas' later (highly aggrandizing) account of Columbus, and since Las Casas wasn't there with King João, there's a very good chance that this story was just invented whole-cloth. But the sentiment is largely true: Portugal scrambled to salvage its mistake by appealing to existing treaties about Atlantic territory, claiming that they had a right to whatever Columbus had found. The Treaty of Tordesillas would later rule in favor of Spain, but the Portuguese fought tooth and nail to get their hands on Columbus' findings. Describing the crown's reaction as "severe regret" would probably not be an overstatement.
When it comes to Spain, I don't personally know of any specific documents that record the reactions of Columbus' royal detractors, like Hernando de Talavera. I can, however, point to some of the strategies they probably used to rationalize the shocking discovery.
Columbus is known for basing his plans on an underestimation of the Earth's circumference, an opinion that was not mainstream before or after his discoveries. However, there was another geographic uncertainty -- and a more controversial one -- that also came into play. Geographers debated how far east Asia extended. There were all manner of divergent calculations for this distance, and any geographer trying to come to a conclusion would have to sort through Ptolemy, Marco Polo, etc. Columbus, for obvious reasons, opted to believe that Asia extended very far east; this opinion, more so than his circumference calculation, would have had reputable backers amongst contemporary geographers and ancient sources. After Columbus came back from "Asia," scholars would have been inclined to give the "long Asia" theory more credence. Peter Martyr, for instance, worked in the court of Ferdinand and Isabella, and wrote this in 1493: "No great stretch of sea separates the shores of India from Spain." To buttress this claim he cited the authority of Aristotle and Seneca.
It was also possible to rationalize Columbus' discoveries as a far-flung extension of the Canary Islands. Obviously, this is absurd in light of modern geography. But Spain had a serious motivation to make this claim, because it would ensure that the New World islands fell into Spanish (and not Portuguese) hands. Spain already owned the Canary Islands, so lumping in Columbus' discoveries with them stripped Portugal of any possible claims. When the Spanish crown initially edited Columbus' letter, they fudged the data to give this impression. Columbus' voyage had taken 33 days from the Canaries, but the edited letter changed it to 20. They skewed the latitudes that Columbus had calculated to match the Canaries exactly, 26 degrees north. Other Spaniards picked up on this idea, especially seizing on the apparent connection between the Canary natives and the Caribbean natives: naked, primitive, without religion. Peter Martyr described Columbus' discovery as part of the long Castilian conquest of the Canaries, a kind of crusade to bring Christianity to these simple people. That Italian rhyming version of Columbus’ letter said something similar in its last line: “the history of the discovery of the new Indian islands of Canary is finished.”
So, at least in the immediate aftermath of Columbus' return, a few things happened: 1) the Spanish crown felt validated in its gamble and tried to manage Columbus’ reports for political gain; 2) Portugal expressed regret at snubbing Columbus and tried to justify their ownership of the new islands; and 3) scholars rationalized the discovery by taking the "long Asia" view or the "long Canaries" view. There's no indication that anyone seriously reconsidered the circumference of the earth based solely on Columbus' success.
It would be in the early 16th century, particularly after the account of Amerigo Vespucci, when people finally saw the Caribbean as part of a "New World."
Sources:
David Abulafia, The Discovery of Mankind: Atlantic Encounters in the Age of Columbus (New Haven: Yale University Press, 2008).
William D. Phillips Jr. and Carla Rahn Phillips, The Worlds of Christopher Columbus (Cambridge: Cambridge University Press, 1992).
Valerie Flint, The Imaginative Landscape of Christopher Columbus (Princeton, NJ: Princeton University Press, 1992).
It's thought that Amerigo Vespucci was one of the first and likely the first to openly suggest that America was a "new world" and not Asia. He postulated this theory while on the coast of modern Senegal, conferring with sailors who had just returned from India by way of Good Hope (Cabral's men). Vespucci himself was part of a crew who was on its way to Brazil to follow up on Cabral's discovery of that place, but had not crossed the Atlantic yet. By chance they convened at Bezeguiche (modern Dakar) at the same time, and it is thought this is where Amerigo, who was familiar with the West Indies and had been there before, first began to think that the India Cabral's men described to him was totally different continent from the one he knew - West Indies and East Indies. He composed a letter while there which first recorded this idea, preceding his more famous letter that got him enough recognition to name two continents after him.
This letter to Lorenzo de Pier Francesco de Medici written 1501 from Bezeguiche in modern Senegal can be found in an English translation in W. B. Greenlee's The voyage of Pedro Alvares Cabral to Brazil and India,1938, page 151.
You might be interested in this letter from Colmbus which is available as audio on librivox here or archive.org has the scanned text.
In this letter printed first in 1493 translated into Latin he describes his discoveries, naming the islands and various fantastic features of the newly discovered lands. It's worth noting this is obviously a work of propaganda and it's hard to say how reliable any of it is.
Follow up question: I've heard that Colombus thought he could get to Asia, not because he thought the earth was smaller in diameter than the then-current consensus, but because he thought the earth was pear shaped instead of spherical.
Is this true?
If so, did it cause any major doubt or revision in the scientific community in Europe after Colombus was successful (and before it was recognized that he didn't reach Asia)?
As a follow up question, how much did Europeans know about the East Indies at the time? Would the appearance and technology/culture of the Americans that Columbus came across been roughly the same as those from the East Indies?