As of 1517 A.D., the Ottmans have acquired the whole Levant, Egypt and they've largely become Muslim. Why they did not move their capital to a historically Islamic city like Damascus, Baghdad, or Cairo? Did any discussion on the topic ever take place?
The Ottoman Empire was not an Arabian empire, but a Turkish one. So they had no reason to uproot their capital from Turkish lands and move it to an Arabian region.
Constantinople Istanbul was way better than Damascus, Cairo, and Baghdad at everything. It had a strategic position on the Bosphorus Strait, and was also close to Europe, which allowed for better coordination of war and trade. It was the crown jewel of the Eastern Roman Empire in terms of culture and wealth, and it was even close to the previous Ottoman capital, Edirne. It was well-defended with two sets of famous walls.
Location: it is a strategic point, the central point for which Black Sea trade must pass. If you are under siege and you have a fleet, you can get supplied indefinitely; if the enemy assaults the walls you can flee to the Asiatic mainland.
To illustrate this point, think how much did the Byzantine Empire did resist while being just that city. And even then, the final conquest was not easy.
Infrastructure: The Byzantines did a lot of work in the city. From Hagia Sophia to the walls. It also had well developed merchant colonies that ensured a nice income. After the sack of Baghdad, there were few comparable cities in the entire world, and none of those under Ottoman control.
So, you leave the city to any governor, and suddenly that governor acquires a lot of power and wealth.
Turkish is not Arab!! While both were Muslims, they certainly did not feel part of the same group. Ottoman conquests in Egypt and Persia1 were as bloody and contested as any of their European campaigns agains Christians.
Certainly, you do not want to put your throne in a city far away from your power base. A well-timed (or badly-timed, if you are the ruler) insurrection and your head is at the stake.
1Which by the way were not Arabs, either.
Two more factors: The Ottoman Turks considered themselves to be the successors of the Roman Empire. Before they captured Constantinople, their territory was known as the Sultanate of Rum. After capturing Constantinople, Mehmed II called himself the Caesar of Rome (Kayser-i Rum). Constantinople had been the capital of the Roman Empire for more than a thousand years. So moving the capital would amount to abandoning that claim.
Although the Balkans had not been ruled for long by Moslems, they were better watered and hence more agriculturally productive and valuable than most of the Arab world. So it made sense to have a capital near that area.
The Ottoman Empire saw itself as a blend of Western and Eastern history. Most of the Ottoman sultan's ambitions were to be the "Alexander the Great" (Check Süleyman the Magnificent). Ottoman rule and ideology was somewhat based on this view point.
The Ottoman Empire never had the ambition to expand eastward. The main expansion was done westward, Italy being the main source of attraction. After the reign of Yavuz Sultan Selim, eastern expansion was done in 6 years - which changed the viewpoint of the empire from being a western renaissance one to an eastern-western mix.
This mix was always repeated in various important ferman. Check the Tanzimat Fermanı for the importance given to this way of life. Also you need to check Namık Kemal, who was a key figure in formation of the ottoman constitution. Check the ideology of Ottomanism too.
After the conquest of Istanbul, probably the most important figure of renaissance, Fatih Sultan Mehmet, looked westward and had ambitions towards Italy. He was the most popular figure in Italy back then, even people of Rome established coins in Latin in his name. Ottoman science flourished in his reign, most of the Latin/Greek books were translated to Arabic (not Turkish since Arabic is considered as the language of science). However, after his reign, this western concentration turned into the west-eastern blend as I've told above.
Why move the capital to Damascus? Has nothing to do with the empire's ambitions...
Apparently these ambitions HAVE failed (as it failed in Hellenistic times), and the founder of the Turkish Republic moved away from east and turned their faces towards west. Now this is how Turkey has been for some time now (almost 100 years).
Basically, the Ottomans were "Turks" first, and Moslems second. Hence, they felt no need to cater to their Arab subjects by adopting one of their cities as a capital in a way that the Mongols might have felt with say, the Chinese. Even today, "Turkey" is not really part of the "Arab" world, even though they are both Moslem peoples.
Besides its political implications (e.g. as the "second Rome"), Constantinople was an ideal capital for the Ottoman Empire. That's because it had aspirations to the north, as well as the south. The fact that the Arabs had been conquered earlier was "incidental;" again, unlike the Mongols, the Ottomans were not headed in a primarily southerly direction. The Ottomans had interests in the Balkans, and also took the Crimean Khanate under its wing, thereby involving itself in "Russian" politics until the twentieth century.
Also, Constantinople guarded one of the three key "straits (the Dardanelles) in the Mediterranean. The other two were the Strait of Gibraltar (outside Ottoman domains) and the Suez Canal (not built until the late 19th century). Apart from Constantinople's strategic location, it was easy to defend, largely protected by water (as the British found out at Gallipoli in 1915), and accessible by land only through narrow corridors.
The main reason as to why the Ottomans never established (or "moved") their imperial Capital to an Arab Capital-(i.e., such as Damascus, Cairo or Baghdad), is because the Ottomans were of ethnic Turkish descent and were not of ethnic Arab descent.
When the Ottomans came on the historical stage, they effectively replaced the 1000 years of Arabian based Islam, with a Turkic style of Islam.....in the conquered (once Christian) city of Constantinople.
When you look at the political and strategic geography of the above mentioned Arab Capitals, versus Constantinople, you will notice that with the possible exception of Cairo (and its proximity to the Nile River), both Damascus and Baghdad were distant from major waterways-(which in early Modern times, would have been quite significant both in terms of commercial and naval supremacy).
However, the city of Constantinople, was (and is still), a city that was (and is still), blessed with a number of key waterways, including, the Dardanelles, the Bosporus and its close proximity to the Black Sea, as well as its entry way into the Aegean and greater Mediterranean sea. One of the main reasons as to why the Byzantine Empire lasted for 1000 years, was greatly attributable to the geopolitical and strategic value of Constantinople. The Turks understood this and spent years penetrating and chipping away at the historic City, aiming to make it the new Capital of their Empire and even the new Epicenter of Islam-(with the notable exceptions of Mecca and Medina in Arabia).
Comparatively speaking, the Caliphate cities of Damascus, Baghdad, Cairo (and even Cordoba in Andalusia, Spain), did not have the same historical longevity and legacy of Byzantine Constantinople, because each of these cities had their own respective Dynasties and with it, frequent feuds and struggles for Power. There was really no centuries long concentration of Power in a single city during the Medieval Arab Islamic period....until the arrival of the Ottoman Turks into Constantinople during the mid 1400's. Like the Byzantines before them, The Ottomans never relocated their Empire-(the relocation of Turkey's Capital took place during the immediate aftermath of World War I, whereby the central Anatolian city of Ankara, became the new seat of Turkish rule, which lasts into the present-day). And in doing so, the Ottomans were able to effectively govern a tri-continental power that was largely uninterrupted for nearly 500 consecutive years.