This is one of those questions that is so broad, it could hardly be answered in the context of a book, let alone a short post. I have written a long post elsewhere trying to lay out the basic attitude of representative Church figures from the 5th to the 13th centuries here. In this I argue that the conscious relationship between the Church and Science is one of ambivalence, that is to say, it has in the broadest possible sense, the Church's position that science is not a matter of direct interest for it and as such much of the relationship has been unintentional.

However, this certainly an overly simplistic picture (as I note in that thread too). There were important (and sometimes conflicting) interests within the Church around the development of science, although they cannot be adequately characterised as interests of the Church as such and they remain more contextually contingent. There are two points that are worth noting right away that play into this interrelationship. On the one hand, although overall we may characterise the Church's view towards science as ambivalent, there is an important exception here, and this insofar as science is seen to conflict with doctrine. Now what exactly this means, and how it plays out are very complicated questions that can certainly not be summarised for the history of the Church as a whole. The complications here are nicely illustrated by the tribulations around the use of Aristotle in the universities in the 13th century.

With the reintroduction of Aristotle in the west, with the translation movement that was well underway by the late 12th century and essentially complete by the mid-13th, at least so far as Aristotle is concerned (for anyone interested there is a good list of translation dates in The Cambridge History of Later Medieval Philosophy, 74ff., though it may be slightly dated at this point). By the early thirteenth century, this was already seen as infringing on the on maters of theological doctrine. At stake here were issues like the creation vs. eternity of the world (you can read Siger of Brabant's treatise thereon here if you're interested), theories of the heavens, the agent intellect, the bounds of God's power, etc. (You can have a look at Giles of Rome's Errores philosophorum for a taste of the sorts of things people were concerned about.) This resulted in a long string of condemnations of Aristotle at the university of Paris, starting in 1210. Although, these were, so far as we can tell, utterly ineffectual and by 1255 the entire Aristotelean corpus was on required reading for students at Paris. Nevertheless, they continued to condemn various Aristotelean theses, most famously in 1277, when Etienne Tempier, Bishop of Paris, condemned 219 theses by members of both the Arts and Theology faculties at Paris, charging them generally with holding the so-called idea of double truth, noting of these scholars in the preamble to the list of theses: "For they say that these things are true according to philosophy but not according to the Catholic faith, as if there were two contrary truths and as if the truth of Sacred Scripture were contradicted by the truth in the sayings of the accursed pagans" (A selection of the condemnations is translated here). This is a particular high point in a struggle around the relationship of faith and reason triggered by engagement with the Aritotelean corpus. I won't go into great detail here as there is an excellent article on the SEP by one of the leading scholars on the matter. But broadly, this is an interesting and much disputed event, as since Duhem's monumental history of science Le Système du Monde. Histoire des Doctrines Cosmologiques de Platon à Copernic (for the early controversy around this see here, though note that this is a somewhat polemical take on the affair). However certainly since the 1960s this event has been taken very seriously and it has been argued that it broke the thrall of dogmatic aristoteleanism, making the way for the development of empirical sciences as we now know them. (With the proceeding century certainly being the most fruitful in the context of medieval developments in science.)

This is a useful illustration of some of the complications involved in considering the churches role in the history of science, as it illustrates not only the way that it at various points actively intended to curtail "scientific" speculation, yet at the same time, this is fundamentally not about science and indeed arguably had significant positive implications for the development of science.

Since you bring up Galileo, it is worth saying a few words about him here. Here is a recent thread about Galileo that covers the facts quite well. However, as I noted in my comment in that thread (which was perhaps a tad more aggressive than it should have been) I have serious reservations about the interpretation of the top post, as it is based heavily on a Catholic apologetic account of the events which is not generally accepted in the mainstream literature. For a good mainstream account of the events, have a look at this article by Olaf Pedersen and to fit this into a broader academic response to the conflict these see this article by David Lindberg and Ronald Numbers. However, not adequately covered in any of these is the negative implications of the Galileo trial for science, a point which we shouldn't loose sight. On this point see particularly William Ashwood's "Catholicism and Early Modern Science" in Lindberg and Numbers (eds.) God and Nature, where he does a good job of showing how the Galileo trial created a serious chill on various sorts of scientific activity in Italy for the remainder of the 17th century.

Finally, on the point of institutions. In the first place, though the Church was deeply involved with the Medieval universities, they weren't owned by the Church. Rather they were corporations of masters (or in the case of Bologna students) that were more akin to a guild than something like a Cathedral school. Just to run with the example of Paris, it developed out a hodgepodge of schools, both religious schools like the Monastery school at St-Victor and Cathedral school at Notre Dame, as well as schools set up by various masters, like Abelard's school. This is not to downplay the impact of the church in their establishment, however it is to note that Universities were as much a result of large groups of tenacious scholars interested in pursuing education and knowledge (again, there were lots of motivations that went into these schools which were emerging at a particularly tumultuous period in western history). For a early history here see Stephen C. Ferruolo, The Origins of the University: The Schools of Paris and their Critics 1100–1215 and Richard Southern, Scholastic Humanism and the Unification of Europe (2 vols) as well as Olaf Pedersen, The First Universities: Studium Generale and the Origins of University Education in Europe more generally. I have myself drawn most of the material for this small section from the chapter on universities in The Cambridge History of Medieval Philosophy.

So where does this get us on your broader question? Can a case be made for or against the Catholic Church (or Christianity more generally) supporting science?

The obvious answer would be yes to both (as the historiography on this point would certainly show), but more seriously the mainstream scholarship seems to be more consistently showing that this is just an unhelpful way to approach the issue in the first place!

On this point I may leave you with a short article by another historian of science, John Hedley Brooke, call aptly A Call for Complexity. (I realise it isn't an academic article, but you can also read the Lindberg & Numbers article which argues substantially the same thing.)

Beyond the material I've reference or linked to above, this material is covered quite well in a number of major Cambridge compilations (mostly noted in my previous post). Notably:

David C. Lindberg and Ronald L. Numbers, eds., The Cambridge History of Science, 7 vols. (Cambridge: Cambridge University Press, 2002-13) Note particularly the articles in volumes 2 and 3 on religion/the church and science in the medieval and early modern periods.

Peter Harrison, ed., The Cambridge Companion to Science and Religion (Cambridge: Cambridge University Press, 2010)

Robert Pasnau and Christina van Dyke, eds., The Cambridge History of Medieval Philosophy, 2^nd ed. (Cambridge: Cambridge University Press, 2014)

Peter Machamer, ed., The Cambridge Companion to Galileo. (Cambridge: Cambridge University Press, 1998).

Norman Kretzmann, et al., eds., The Cambridge History of Later Medieval Philosophy (Cambridge: Cambridge University Press, 1982).

Also: David C. Lindberg and Ronald L. Numbers, eds., God and Nature: Historical Essays on the Encounter between Christianity and Science (Berkeley: University of California Press, 1986).

David C. Lindberg and Ronald L. Numbers, When Science and Christianity Meet (Chicago: University of Chicago Press, 2003).

Edit: finis litterae causa