Well, no, actually. It is true that modern science has demonstrated that the universe is incomparably larger than the premoderns believed. But this is not the same thing as showing that they believed the Earth to be the largest thing in the universe, much less that the universe itself is small.
An excellent book on this is Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley by Albert Van Helden. He goes into some detail with the specific calculations given by ancient and medieval cosmologists, so if you want more detail it's an excellent resource. Unfortunately only the first 40 pages or so are given to ancient and medieval views. Another author who comments on this, specifically as it touches on Christianity, is C.S. Lewis in The Discarded Image, as well as chapter 7 of Miracles. Interestingly (or ironically, depending on your point of view), Van Helden takes issue with Lewis regarding Roger Bacon's Opus Maius. I think it's based on a misunderstanding, but I won't go into it here. (Update: I address their disagreement in an addendum.)
As a simple matter of fact, the ancients and medievals believed that the universe was larger that we can imagine, and that the Earth should be considered a mathematical point, infinitely small, within it. In the fourth century BC Aristotle, Eudoxus, and Calippus argued for a spherical Earth, and this was the consensus model thereafter. The significance of this is, as Van Helden argues, "One of the postulates of spherical astronomy is that the Earth can be considered a mere point in relation to the spheres of the heavenly bodies." This belief allowed them to assume "that the Sun's rays striking the Earth are parallel, even at locations far removed from each other" on the Earth's surface. Aristotle, for example, wrote in De Caelo 2:14 that "our observations of the stars make it evident, not only that the earth is circular, but also that it is a circle of no great size," pointing out that even a small change of location on Earth (from Greece to Cyprus to Egypt) results in different stars being seen. Others followed this belief, such as Eratosthenes (third century BC), Hipparchus (second century BC), and finally Ptolemy (second century AD) whose system became the accepted cosmological model in the Middle Ages. Ptolemy specifically wrote in Almagest 1:5 that, "The Earth, in relation to the distance of the fixed stars, has no appreciable size and must be treated as a mathematical point."
In Planetary Hypotheses, Ptolemy used the mathematical models and parallaxes he had calculated in Almagest to determine the sizes and distances of the planets and the stars. In determining the distances of the planets, Ptolemy employed the Aristotelian doctrine that there are no empty spaces between spheres, known as the "nesting spheres" framework. As such, Van Helden states, "The greatest geocentric distance of one planet therefore had to equal the least distance of the next higher planet." Thus the furthest distance of the last planet (Saturn) equalled the distance to the sphere of fixed stars. Using this, Ptolemy calculated the distance to the sphere of stars as 19,865 earth radii, which translates to approximately 80 million miles or 130 million kilometers (using Eratosthenes' measurement for the Earth's radius). In fact, Ptolemy states that this may only be the minimum distance: "if all the distances have been given correctly, the volumes are also in accord with what we have said. If the distances are greater than those we described, then these sizes are the minimum values possible."
Ptolemy's system completely dominated astronomy until the Modern Age. According to Van Helden:
From the second to the sixteenth century, astronomy was a commentary on Ptolemy. No man ever wielded posthumously such a pervasive and long-lived authority in astronomy, and it is to be doubted that anyone ever will again. Ptolemy's work superseded the efforts of all his predecessors -- surely one of the main reasons why so few of their works have survived -- and it defined the astronomical problems for his successors, at least until the time of Tycho Brahe and Johannes Kepler.
In the sixth century AD, Boethius had the character Philosophia in The Consolation of Philosophy (one of the most widely-read books throughout the Middle Ages) tell him that the Earth is so small and the universe so large that the former should be treated as a mathematical point. Almagest was translated into Arabic three times in the ninth century AD, and Planetary Hypotheses was as well (only once though). Among the many Muslim commentators were al-Farghānī, Thābit ibn Qurra, and al-Battānī. The Christian West was heavily influenced by the Muslim astronomers, and their works (as well as Ptolemy's) were translated into Latin, beginning in the twelfth century. Van Helden writes, "it is fair to say that virtually all educated persons after about 1250 were familiar with the principle of nesting spheres and the cosmic dimensions derived from it." As with the Muslims, the Christians accepted Ptolemy's cosmic distances, only fine-tuning them here and there. Campanus, for example, gave the distance to the sphere of the stars as 22,612 earth radii, which translates to over 73 million miles using a more accurate measurement of the Earth's radius as 3,245 miles. One of the most prominent scientists to comment on Ptolemy was Roger Bacon. He gave the distances of the astronomical objects in miles, putting the distance to the sphere of fixed stars at 65,357,500 miles. Since this is essentially the radius of a sphere, Bacon doubled it to reach a diameter of 130,715,000 miles, and multiplied this by pi to reach the universe's circumference at 410,818,517 miles (and three-sevenths).
Ptolemy's astronomy was firmly embedded in medieval society. According to Van Helden, "the Ptolemaic cosmic dimensions can be found throughout the spectrum of the literature of the High Middle Ages, from the technical to the popular." As such, it obviously exerted influence in other genres of writing. He cites the thirteenth century French poem Image du Monde, The South English Legendary from the same century (also cited by Lewis), and Dante's Convivio. Lewis writes in Miracles
More than seventeen hundred years ago Ptolemy taught that in relation to the distance of the fixed stars the whole Earth must be regarded as a point with no magnitude. His astronomical system was universally accepted in the Dark and Middle Ages. The insignificance of Earth was as much a commonplace to Boethius, King Alfred, Dante, and Chaucer as it is to Mr. H. G. Wells or Professor Haldane. Statements to the contrary in modern books are due to ignorance.
The real question is quite different from what we commonly suppose. The real question is why the spatial insignificance of Earth, after being asserted by Christian philosophers, sung by Christian poets, and commented on by Christian moralists for some fifteen centuries, without the slightest suspicion that it conflicted with their theology, should suddenly in quite modern times have been set up as a stock argument against Christianity and enjoyed, in that capacity, a brilliant career.
So why are we so willing to believe the premoderns thought the Earth was the largest thing in a small universe? Part of it, no doubt, is because we compare their conception of the universe's size with our own and recognize that we know the universe to be unimaginably larger than they thought it to be. What this fails to recognize, however, is that they were starting with an unimaginable size. The universe was, to the premoderns, larger than we can fathom. Multiplying an unfathomable size by a thousand, a million, or even by another unfathomable size, yields ... an unfathomable size. The specific differences can be mathematically expressed of course, and they have a great deal of value for our understanding of the universe. But as far as the human imagination is concerned, the modern "discovery" that the universe is larger than we can imagine was something everyone already knew. As Lewis writes in Discarded Image, "the fact that the height of the stars in the medieval astronomy is very small compared with their distance in the modern, will turn out not to have the kind of importance you anticipated. For thought and imagination, ten million miles and a thousand million are much the same. Both can be conceived (that is, we can do sums with both) and neither can be imagined; and the more imagination we have the better we shall know this."
However, Lewis argues, there is a possible counter-argument to all of this. Many times in premodern literature, characters are taken outside of Earth to the sphere of the Moon, or even of the fixed stars. From this vantage point, they then look down upon Earth, and see all kinds of details which would be impossible to see from a great distance. Doesn't this suggest that they did not really perceive the distances to be very great? Lewis's answer:
The impossibility, under the supposed conditions, of such visual experiences is obvious to us because we have grown up from childhood under the influence of pictures that aimed at the maximum of illusion and strictly observed the laws of perspective. We are mistaken if we suppose that mere commonsense, without any such training, will enable men to see an imaginary scene, or even to see the world they are living in, as we all see it today. Medieval art was deficient in perspective, and poetry followed suit. Nature, for Chaucer, is all foreground; we never get a landscape. And neither poets nor artists were much interested in the strict illusionism of later periods. The relative size of objects in the visible arts is determined more by the emphasis the artist wishes to lay upon them than by their sizes in the real world or by their distance. Whatever details we are meant to see will be shown whether they would really be visible or not. I believe Dante would have been quite capable of knowing that he could not have seen Asia and Cadiz from the stellatum and nevertheless putting them in. Centuries later Milton makes Raphael look down from the gate of Heaven, that is, from a point outside the whole sidereal universe -- 'distance inexpressible By Numbers that have name' (VIII, 113) -- and see not only Earth, not only continents on Earth, not only Eden, but cedar trees (V, 257-61).
Thus, these examples do not demonstrate that the ancients and medievals thought the universe was small, or even that its size was imaginable. On the contrary, they recognized that the universe was larger than we can fathom, and that the Earth was, for all practical purposes, an infinitely small point within it.
Another possible counter-argument might be that, even if the premoderns clearly believed the universe to be unimaginably large, they still believed the Earth to be the largest -- and therefore most important -- thing in it. But this is simply false. Aristotle argued in De Caelo that "compared with the stars it [the Earth] is not of great size." Lewis points out that Cicero, in Somnium Scipionis, recognized "that the stars were globes which easily outstripped the Earth in size. ... This passage was constantly in the minds of succeeding writers. The insignificance (by cosmic standards) of the Earth became as much a commonplace to the medieval, as to the modern, thinker; it was part of the moralists' stock-in-trade, used, as Cicero uses it (xix), to mortify human ambition." Besides Cicero, Lewis also lists Chalcidius and Macrobius as believing, "like everyone else," that the Earth was smaller than the smallest star, and argues that this belief was held throughout the Middle Ages. According to the Ptolemaic system, only Venus, Mercury, and the Moon were smaller than the Earth (which, in fact, they are); everything else was larger. With Ptolemy's system achieving near universal assent in the Middle Ages, these sizes almost never varied.
In fact, even if we ignore this -- even if we assume for the sake of argument that the premoderns thought the Earth was larger than the stars, the planets, and the Sun -- we still cannot ascribe any significance to it based on this. On this view, the Earth would still not be the largest object: "The furthest sphere, Dante's maggior corpo is, quite simply and finally, the largest object in existence." In their cosmology, there was no such thing as empty space; all of the vast distances between the stars and planets were completely filled, and as such, each sphere constituted an object of overwhelming size. In comparison to the most distant sphere, that of the fixed stars, the Earth was, as far as our imaginations are concerned, infinitely small.
Thus, the claim that the premoderns believed the Earth and its inhabitants significant because it was the largest thing in a small universe doesn't even get off the ground. They thought the Earth was one of the smallest things in an unfathomably large universe. While modern science has certainly corrected their cosmology on many points, it has not altered this part of the picture. If they did think an object's significance was related to its size, then they would have concluded that the Earth is one of the least important, significant, or valuable places in the universe.
Update (11 Aug): (see also part 1 and part 3)
(cross-posted at Quodlibeta)