Is there any evidence that the Catholic Church slowed innovation during the Middle Ages?

To answer this question the first step is to define innovation. As you quote Greeks and Romans I'll assume you mean innovation in a broad sense as the expression of new thoughts as this what those civilizations are most famous for.

Then what actions did the church take on this topic? The Middle Ages are a thousand-year-long period and the situation changed a lot from the early centuries to the last ones. The Dark Ages were difficult times for most of Europe and a lot of what once was went into oblivion. Then the Church actively protected ancient texts and monks kept translating them. However this can't be considered innovation, it shows an interest in the transmission of culture that is essential to innovation in itself. The machine was just not ready to start.

When the situation got better, and Church actions have a lot to do with it, the times were favorable for a rebirth in intellectual activities. This is when Universities were born. There was a great ferment in culture in the first centuries of the new millennium: we can see pan-European cultural exchanges and the first protagonists were all members of the clergy, as it was the only way that people could dedicate themselves to thinking and travel. So the Church is the original reason intellectuals existed. Of course, the main topic discussed was religion as this was what interested people at that time. New ideas were developed and even if sometimes there was fierce resistance (even St Thomas had some hard times) they weren't rejected a priori. Literature and the arts flourished and evolved. The variety in written texts' styles shows the freedom of the writers, that were not bound by any rules other than their and their society's taste and culture. That's how new ways, that didn't exist in Antiquity, of doing both poetry and prose were born. Ironically Renaissance actually represented a step back on this. As for the Church involvement in all this suffice to say that if it wanted it could have easily struck down these evolutions. On the contrary thinkers could often rely on Church revenue: famous examples are Boccaccio and Petrarca.

If you want to focus on creativity applied to material purposes it is once again in the church's ranks that you will find the most, with monks being the prime source: following the rule "ora et labora" they dedicated their skills and knowledge to manual works, resulting in many inventions, ranging from tools to alcohol (champagne for example...). It is true though that the Church tried to ban some of the instruments that appeared in that time: a famous example is the one of the crossbow, declared illegal against other christians by the Latran council of 1139. The reason for this is the horrifying murderous power that this weapon gives to any wielder, even an untrained one. This prohibition may be the kind of obstacles to innovation the question was looking for but it can also be seen as a mark of civilization, not unlike the current ban on chemical weapons, that are now forbidden for similar reasons.

Overall the Church had both a role as a suppressor of what was immoral, which in perspective can be interpreted more as a protection of the population from this evil; and a role as a stimulator for development. However, if we cannot see great technological or scientific leaps during the Medieval times, the greatest impact of the Church on innovation was simply making it possible by creating the right mindset. It is certain that the Occidental way of thinking, that was forged by Christian values, was what determined that it was this people to bring about the greatest revolutions in science and technology.

Well, it depends what you mean by, "rate of innovation". It is true that the Ancient Greeks were intellectually innovative and helped to broaden many of the disciplines we study in the present-day. And it also true that the Romans were architecturally innovative, as well as very innovative in civil engineering and what we today call, "urban planning".-(One COULD make the case that the city of Rome, 2000 years ago, was the 1st major city in World History).

With the collapse of the Western Roman Empire in 476 AD/CE, "The Dark Ages" ensued and much of the Italian peninsula, as well as most of Northern and Western Europe, were at a cultural standstill-(or even regressing towards a more primitive and unsophisticated age) for the next 575 years. The only notable exception of cultural advancement during this time, was the short lived and regionally based, Carolingian Renaissance ushered in by Charlemagne at his Capital in Aachen in the Northwest corner of Germany during the late 700's AD/ CE. Aside from the Carolingian Renaissance, "Dark Ages" (Northern and Western) Europe, as well as much of the Italian peninsula, saw no major intellectual, cultural or political innovations.

One could make the case that with the emergence of the Late Middle Ages, (beginning in 1050 AD/CE and ending in 1400 AD/CE), much of the Italian peninsula, as well as Northern and Western Europe, began to see innovations in University education, architecture, the translation of classical works, as well as the refinement of cities and towns. However, the European Catholic Late Middle Ages were still largely deficient in the sciences, medicine, as well as technology and such "innovation" in these fields, would come with The Northern Italian Renaissance, followed by the Scientific Age and The Enlightenment, generations and even centuries later.

The common connection to the Early and Late Middle Ages in much of Europe, was the Catholic Church. As an institution, the Early Medieval Catholic Church, was ultra-religious initially, distrustful-(perhaps even contemptuous) of anything that resembled paganism, including the scientific and intellectual achievements of the Ancient Greeks. For the Early Medieval Catholic Church, the emphasis was on Christian religiosity/piety; the values and societal norms had to strictly adhere to a type of Dogmatic Catholicism, which did not include, advancements, refinements or "innovations" in thought.

So yes, the Early Medieval Catholic Church did indeed slow down or decelerate innovations in thought throughout much of the European continent-(the aforementioned Carolingian Renaissance being the noted exception). This would change during the Late Middle Ages, though the change was not so dramatic and the innovations in the sciences, medicine and technology were still largely absent from this part of the world during this time period. Again, the Catholic Church, due to it near indomitable presence, was the primary obstacle towards such progress and "innovation".

Hypatia was killed by order of the Church for being a scientist who dared to believe things that were contrary to Holy Scripture... and so was Giordano Bruno. In between, there were a lot of people killed and documents declared heretic.

Then there is the Index Liborum Prohibitorum - an Index of Prohibited Books - that made scientific texts like Kepeler's unavailable in Catholic countries from the creation of the printing press until well into the 18th century.

Let's look at some of the scientists banned by the Index:

• Pascal
• Bacon
• Copernicus
• Galileo
• Hume

So, there's some direct evidence, altho the Index is more early modern than medieval.

The greatest destruction of knowledge caused by Christianity occurred much earlier at the start of the dark ages. The burning of the library at Alexandria and the murder of Hypatia (she was skinned alive with oyster shells) is but one famous example of the damage wrought at that time by christian mobs. These were made up of people who expected the world to end any day and who had often given away all their possessions to follow some wandering preacher. They moved across the countryside like a plague of locusts and were very destructive. As most couldn't read they burned pretty much anything written just in case it might be heretical and killed anyone who could write in case they might be a heretic or witch. The church ironically was responsible for saving much written material from these mobs as only they had the authority to turn the mobs away. While the church was mostly concerned with saving records of early church history they saved many other documents in the process.

In the middle ages the church was more interested in controlling and suppressing knowledge rather than outright destroying it. Ironically Galileo appeared towards the end of this period when knowledge was starting to escape from church control.

The Catholic Church, especially in the High Middle ages, was a great aid to science. The Church even founded the modern university system.

Physicists of the High Middle ages had such a profound affect on the intellectual atmosphere of Galileo, Newton, et al. that they took their discoveries as common knowledge. Some of the most famous of the pre-Galilean physicists were:

• The French bishop, physicist, and economist Nicole Oresme, who determined the mean speed theorem of uniformly accelerated bodies: v_avg = v_f / 2.

• Bishop Oresme posed the famous Gedankenexperiment:

  I posit that the Earth is pierced clear through and that we can see through a great hole farther and farther right up to the other end where the antipodes [poles] would be if the whole of this Earth were inhabited; I say, first of all, that if we dropped a stone through this hole, it would fall and pass beyond the center of the earth, going straight on toward the other side for a certain limited distance, and that then it would turn back going beyond the center on this side of the Earth; afterward, it would fall back again, going beyond the center but not so far as before; it would go and come this way several times with a reduction of its reflex motions until finally it would come to rest as the center of the Earth....
  
  {Quoted by K. V. Magruder from Le Livre du Ciel et due Monde (Madison: University of Wisconsin Press, 1968), translated by D. Menut, pg. 573.}

• Bishop Oresme wrote (before Galilean relativity): “If air were enclosed in a moving ship, it would seem to the person situated in this air that it was not moved.” Book of the Heavens, Book II chapter 25, from Grant, A Source Book of Medieval Science, pg. 505, Harvard, 1974

• Jean Buridan (d. ca. 1359) invented/discovered the concept of momentum and the equation p = m × v.

• Thomas of Bradwardine (c. 1295-1349) distinguished mean and instantaneous velocity.

• Bradwardine determined in 1300 that for uniformly accelerated objects, $d = \frac{1}{2} a t^2$, which De Soto, O.P., (b. ca. 1494) applied to free-falling objects; Bradwardine thus wrote the first physics equation.

• Jordanus de Nemore and Torricelli influenced Galileo's treatment of inclined planes.

Robert Grosseteste (c. 1168-1253) did experiments (not yet of course with modern rigor) and was keen on using mathematics; he is known for his work on understanding the rainbow. Thomas of Bradwardine (c. 1295-1349) at Merton College Oxford introduced the distinction between mean velocity (x/t) and instantaneous velocity (dx/dt) [and he was the first to write a physics equation]. Bradwardine had an enthusiasm for empiriometric physics that started a whole school called the Merton school (his successors include: William Heytesbury, Richard Swineshead, and John Dumbleton) that was extremely influential throughout Europe. Among other things, they were known for the Merton mean speed theorem, by which they proved the correct formula for free fall distance was given by s = ½ a t². Interestingly, both Bradwardine and Grosseteste at some point in their lives were Archbishops of Canterbury. Nicole Oresme (<1348-1382) and Giovanni di Casali (c. 1350) independently developed use of 2-D graphs [long before Descartes (1596-1650)]. Oresme described all change using these graphs in particular local motion, including calculating area (integrating) under velocity curves to get distance. Oresme's arguments for the sun-centered and moving earth were widely known: he said, for example, that "...not only is the earth so moved diurnally, but with it the water and the air, as was said, in such a way that the water and the lower air are moved differently than they are by winds and other causes. It is like this situation If air were enclosed in a moving ship, it would seem to the person situated in this air that it was not moved." (p. 133, Dales.)

—A. Rizzi's Science Before Science pgs. 199-200

Even further back, Philoponus (late 5th, 2nd ½ of 6th century A.D.; also called "The Grammarian" or "The Christian") is impressive:

He argued that the sun is fire and of terrestrial-like, corruptible matter. He devised a precursor to the notion of impetus which Buridan later developed, that which keeps moving bodies in motion even after the mover ceases being in contact with them; air does not keep projectiles in motion. He discovered that light rays travel the same both backwards and forwards. He invented functions of variables and their "courses" (what we'd call "first derivatives" in modern calculus). He discovered the law of inertia, that bodies in motion remain in motion unless something impedes their movement, literally a thousand years before Galileo, Newton, et al.!

He's certainly one of the "grands génies de l'Antiquité" ("great geniuses of Antiquity") and "principaux précurseurs de la Science moderne" ("principle precursers to modern Science"), as Pierre Duhem wrote in his magisterial, 10 volume work in the history of medieval physics:

• Duhem, Pierre Maurice Marie. Le système du monde: Histoire des doctrines cosmologiques de Platon à Copernic. 10 vols. Paris: A. Hermann, 1913-1959.

Partially translated in:

• Duhem, Pierre Maurice Marie. Medieval Cosmology: Theories of Infinity, Place, Time, Void, and the Plurality of Worlds. Edited and translated by Roger Ariew. Chicago: University of Chicago Press, 1985.

cf. also:

• Hannam, James. God’s Philosophers: How the Medieval World Laid the Foundations of Modern Science. London: Icon Books, 2009.

The Middle Ages reduced innovation in the Middle Ages. If anything, the Church was the only thing keeping Western Europe literate during that time (and even then, just barely). The fact that any books from earlier times survived at all we have Church scribes to thank for.

Knowledge and the dissemination of knowledge is the lifeblood of what you would call "innovation". They were certianly repressive about a lot of things, and certianly could have been more free and open about information, but without the Catholic Church all literary activity in Europe, including the preservation of old knowledge, would have ceased completely.

Quite the contrary as Rodney Stark pointed out in The Victory of Reason - the Catholic church itself promoted most of the societal conditions that allowed the Middle Class to take hold, and in so doing also promote the nurture of science and industry.

Chief among these were personal property rights (stemming from the idea that we were God's stewards) and equality before God in all judicial matters (hence the rule of law). The ideas for patent and copyright also come out of respect for the Imago Dei in all men.

Stark also points out the explicitly Christian roots of the Renaissance, which stemmed from a religious effort to go ad fontes - back to the sources - first of Christianity (hence the glorification of Rome), then to the society in which Christianity flourished.

Stark spends a great deal of time talking about the factors that eventually led Southern Europe (Catholic) to eventually decline after the Reformation - and these mostly boil down to the Roman Catholic church siding with strong men who could protect the church over against the very ideals for which the church stood.

It's a good read and will make the case very persuasively.

Also, along these lines is How the Irish Saved Civilization by Thomas Cahill. There the thesis is that if it were not for the church in Ireland, we would have lost most of the learning and literature of Rome.

If you're looking to make the anti-clerical case, look to Edward Gibbon's polemic Rise and Fall of the Roman Empire. He'll make the case, but most historians agree that it is a polemic moreso than a valid historical argument.