Timur Kuran is a historical political economist, and argues in his book The Long Divergence, as well as the shorter article in the Journal of Economic Perspectives "Why the Middle East is Economically Underdeveloped", (relative to Western Europe and its settler colonies) , argues that there were 3 relatively minor institutional (by institutions, Kuran speaks as economists understand them, as rules--whether they be cultural-normative or legal/formal) differences between the Islamic World and Western Europe that served as "bottlenecks".

1. In Islamic law, inheritance was structured so it was split among children (including women). This prevented the level of capital accumulation that happened in Europe where families' wealth could be kept in the hands of of the oldest--or chosen--son, building up until significant wealth could be invested in economic growth-driving ventures.

2. Islamic law didn't really have mechanisms for making corporations or otherwise pooling wealth from many individuals. You didn't see things like joint-stock companies or other mechanisms for individuals to pool their wealth and make contracts with each other that mitigated the risk involved in handing over money to a joint investor and trusting you'd get some return.

3. The main institution that allowed the prevention of the dispersal of inherited wealth, the waqf, came with strict rules over how that wealth could be deployed. This was a good institution at the time it was enacted, but over time, meant it could not invest in new ways of doing things.

The article is available here http://web.stanford.edu/~avner/Greif_228_2007/Kuran,%202004.%20Why%20the%20Middle%20East%20is%20Economically%20Unverdeveloped.pdf

I can't speak for the law code (strangely enough, given I am a lawyer in real life) or the arts, but I can talk about mathematics and natural philosophy. The TL;DR is: Islamic culture started to shift away from logic to mysticism for reasons I have yet to fully understand, and the organization of the education system may also have been a factor.

The books I just read recently, "Warriors of the Cloisters" (Christopher Beckwith) and "Aladdin's Lamp" (John Freely) both pointed to a single guy, al-Ghazali, who wrote a treatise called "On the Incoherence of the Philosophers". Apparently this work called out all of the differences between Aristotle and Islamic theology/cosmology and was very influential in causing intellectuals in the Islamic world to reject natural philosophy in favour of mysticism.

While I have no doubt that such a treatise could have been very influential, I feel like there have to have been other factors at work. I think it would be unusual for one single work to somehow turn the tide all by itself. Another cultural feature that Beckwith mentioned, though he didn't specifically tie it to the eventual rejection of natural philosophy, was that unlike the Christian West the Islamic lands did not tend to have organized institutions of higher learning that focused on natural philosophy*. Instead, their institutions of higher learning focused on theology and law, leaving any natural philosophers to do their work in relative isolation, and presumably they would have needed either an interested patron or an independent source of income in order to do so. This affects who can do natural philosophy and whether there is anyone to share ideas with, collaborate with, teach, or learn from.

By contrast, when Aristotle reached the West, which it did of course by passing through and being preserved in the Islamic lands, independent institutions of higher learning in the form of universities were also being established. For reasons that have not yet been revealed by my research, these universities did foster the study of natural philosophy (even though the Bishop of Paris tried, several times and unsuccessfully in the 13th century, to ban Aristotle). This meant that, unlike a situation where you have only independent scholars, people can learn from and teach others, collaborate with others, and share their ideas about natural philosophy, which allows a larger body of work to be developed and passed on. In addition, universities are open to more scholars, not just ones who have patrons or an independent source of income.

These are some factors that have been suggested by my recent reading in this area. I feel strongly that there must have been more going on in Islamic culture generally to support the turn away from natural philosophy urged by al-Ghazali, though he was obviously an influential figure.

*There are a few exceptions, like the House of Wisdom in Baghdad, but it was destroyed by Mongols in 1258 and it's said that they threw all of the books in the river, after which the rivers ran as black as ink.

I don’t think there is one single correct reason for the decline OP, as you can probably infer from the number of different answers you’re getting. After all, the Islamic world spans 3 continents, and its decline lasted several centuries depending on the historical narrative you subscribe to. I’m no historian, but I am particularly interested in Egypt’s history during the late Middle Ages. I can talk a little about its decline.

Egypt was (and in many ways still is) the core of the Arabic Islamic world. Following its foundation by the Fatamids, Cairo would rapidly grow to become the premier city in the Islamic world, surpassing Abbasid Baghdad. Egyptian military, economic, and cultural might would reach its zenith under the Ayyubids and the Bahri mamelukes. At the time, Egypt had a large middle class made up of native Egyptian landowners, craftsmen, merchants, and artisans. It had a wildly successful textile and sugar industry; there were over a hundred workshops in Cairo alone. Textiles and sugar were exported to Europe via trade with the Italians. Powerful Karami merchants established large scale business ventures in the spice trade through the Red Sea route. Egypt also controlled the lucrative pilgrim routes to Mecca and Medina. The Al Azhar University would become the most prestigious Islamic fiqh institution in the near east, housing all four madhabi traditions. Egyptian architecture blossomed during this time period; just google pictures of Islamic Cairo and you can see for yourself.

Things begin to change for the worse in the 14th century, coinciding with the Burji Mameluke takeover. The reasons are not clear to me, but the trigger seems to have been the Black Death, from which Egypt never fully recovered. Its population plummeted, and the middle class collapsed. Cairo’s population was estimated at 750,000 before the plague, the reported fatality rate was 60%. The country would also suffer from several outbreaks and famines in the following centuries. When Napoleon landed in Egypt, its population was estimated at about three million. That's down from eight million in the middle ages. Egypt began exporting raw cotton and sugar instead of locally refining, the Karami merchants were increasingly displaced by Italian and Hadrami competitors. The Mameluke elite, faced with plummeting state revenue, increasingly monopolized industry, landownership, and trade. An Egyptian historian of the time, whose name escapes me, reports systematic abuses involving extortion, protection racketeering, and corruption by the Mameluke elite. The Sultanate would become heavily dependent on the Spice trade and its commercial ties with Venice as other sources of revenue declined. So it comes as no surprise that once the Portuguese wrested control of the Indian Ocean after the Battle of Diu and completely blocked trade through the Red Sea, the Sultanate would collapse in the face of the Ottoman onslaught less than a decade later.

This doesn't completely answer your question, but it does show one example of the decline . I’m at work right now, so I'll post sources when I get home.

You're asking an exceptionally complicated question and the answers are naturally going to be varied and politically charged.

I'd credit failure to nationalize as the biggest reason, for a failure of Arabic regions to progress technologically and economically.

There's a Bedouin saying that comes to mind, that translates to "I, against my brothers. I and my brothers against my cousins. I and my brothers and my cousins against the world. "

Arab states are much more tribal than westerners are. The Middle East was divided up by Western agreements after WWI, that made it convenient for the map drawers but not necessarily for the people who lived there. That is to say, the West said to a collection of tribal groups, "You are all Syrians" or "You are all Jordanians" etc.. These people had no reason to be loyal to the collective group. The only way these states hold together is by the strong arm of a dictatorship. Just like when Tito died, when Saddam Hussein was deposed, it lead to massive instability. An environment held together by force of arms with dozens of factions vying for power is not a conducive environment for technological development and economic growth.

There's also the fact that Europe industrialized first (why it did is a whole other debate). Industrialization increases the rate of technological growth. That is when you have the infrastructure to manufacture interchangeable parts parts,it becomes easier to make tools to manufacture parts. It's a feedback cycle leading to economic and technological growth. It also leads to imperialism, as such manufacturing requires resources and industrializing nations became hungry for resources elsewhere. Western powers forced other regions of the globe into exploitative trade deals, that benefited the west (furthering their advantage), but no one else.

"The foreigners chain up Muslims, put around their necks a yoke of servitude, debase them, humiliate their lineage, and they do not mention their name but with insult. Sometimes, they call them savages and sometimes regard them as hard-hearted and cruel and finally consider them insane animals. What a disaster!" - Sayyid Jamal Ad-Din Al-Afghani

Al-Afghani and many of his contemporaries like Liang Qichao from China were quick to realize that the reason the west dominated the rest is because of superior technology and organization. One response to imperialism was "if you can't beat'em, join em". The Tanzimat reforms, were an Ottoman attempt to modernize and retain territorial integrity. Al-Afghani encouraged scientific and philosophic thought, but it was considered too radical and he was ultimately booted out Istanbul. He advocated Arab nationalism saying "There is no happiness except through nationality and no nationality except through language...a people without unity and a people without literature are a people without language. A people without history are a people without glory, and a people will lack history if authorities do not rise among them to protect and revivify the memory of their historical heroes so that they may follow and emulate...all this depends on a national education, which begins with the fatherland, the environment of which is the fatherland, and the end of which is the fatherland."

Japan meanwhile had successfully adopted a nationalistic western style government and industrialized. However, they had embarked on their own imperial ambitions.

So with the collapse of the Ottoman empire after WWI (map drawing spree mandatory), many in the Middle East were looking beyond western models and turned to their own traditions as a source for strength. "Modernization hadn't secured the Ottomans against infidels. On the contrary, it had made them more dependent." -Al Afghani who eventually advocated for a blend of Islam and western scientific institutions.