[deleted]
In the early Medieval period, the Catholic Church was actually responsible for preserving the knowledge and study of science, technology and other learning and knowledge in Western Europe. The structures of the Catholic Church required a level of literacy and numeracy that would have been absent from the vast majority of the population, and resulted in churches and monasteries often containing vast libraries of classical works that would have been otherwise lost. Unfortunately, as the Empire crumbled, so too disappeared any widespread ability to read and write Greek, which had formed the basis for much Classical study, and so often only Latin texts were available.
To be a writer, historian, scientist or chronicler in the medieval period was essentially synonymous with being educated within the Church. Many medieval writers had works called Florelegia which contained select tracts and important works from classical writers, thinkers, philosophers and natural scientists, from which they could quote in their own works. Gerald of Wales, for example, purported to often quote Ovid in his works (although often made things up), and the German bishop Adam of Bremen made use of Pliny the Younger's Historia Naturalis and the Greek writer Ctesias in writing his Gesta Hammaburgensis Ecclesiae Pontificum or Deeds of the Bishops of Hamburg. Mathematics and astronomy were deemed highly important, not least because they were necessary to calculate the ever-changing date of Easter. Natural sciences were held in regard as they were seen to be studying the intricacies of God's creation, thus bringing man closer to the divine.
Although many Classic texts did survive, they were often rare, and there can be said to have been a decline in the study of the sciences during the early medieval period, as the fragmentation of Europe lead to increasing difficulties in accessing texts, but this was is spite of the efforts of the Church rather than because of it. During the course of the Medieval period, however, there were a series of "renaissances" which led to the continued survival and advancement of scientific and technological study. A small one had occurred in ninth- and tenth-century England as a result of Alfred the Great: In England, Alfred the Great was famously a great proponent of education, and according to his biographer Asser required all of his nobility to be functionally literate, or at least raise their children to be able to read, write and perform the basic mathematics necessary to perform the bureaucracy central to the nascent English state. This drive stemmed from Alfred's childhood education in the Church - he was a third son and expected to become a priest rather than a king, and so was highly educated - and the Anglo-Saxon Chronicle and documents such as The Burghal Hidage and the complexities of Æthelstan's c.926 Grately legal codex suggests that this education was passed onto his children. The engineering and design of Æthelflæd's burhs in Mercia was often carried out based on Roman designs, and both she and her father carried out extensive repairs of Roman city defences to a high standard of engineering in cities such as London, Cheshire, Exeter and Bath.
A far larger one occurred in the Twelfth Century, largely as a result of the conquest of Toledo by Christian forces of the Reconquista in Spain, which returned the vast library of Toledo to Christian hands. These contained not only extensive Greek and Latin scientific texts, but also Arabic ones, all of which were translated and copied extensively into Latin on orders of the Papacy and disseminated widely through Europe. Universities such as the University of Paris were founded and soon bloomed as a result of the vast interest in studying these new texts. Many other universities, such as Cambridge and Oxford, soon followed. Scholars such as Gerald of Cremona, who translated over 70 works, were responsible for kickstarting an interest into exploring science, learning and the natural world in order to better understand God's creation. Many learned Greek and Arabic to help translate, and universities encouraged their members to visit other institutions to share and gain knowledge.
Some clergy and theologians were famous in their time as physicists and scientists as much as they were as philosophers and writers. Thomas Aquinas was known as a "Doctor of the Church". Churchmen such as Roger Bacon and Robert Grosseteste studied optics in great depth, alongside theories such as induction, and the natural sciences. Grosseteste founded the Oxford Franciscan College and wrote to the Pope to lobby for increased teaching of natural sciences, and along with Bacon and Aquinas was responsible for establishing many of the principles of the scientific method.
Grant (1996), The Foundations of Modern Science in the Middle Ages
Hannam (2011), The Genesis of Science: How the Christian Middle Ages Launched the Scientific Revolution
Huff (2003), The Rise of Early Modern Science: Islam, China, and the West
Lindberg and Shank (eds.) (2013), The Cambridge History of Science, Vol.2: Medieval Science
This exact question was asked recently and got a great answer from u/restricteddata: https://www.reddit.com/r/AskHistorians/comments/67pwj9/did_the_catholic_church_hold_back_scientific_and/