This is a truly fascinating question with numerous layers. Let's start at the beginning:

It's clear that the early Arab conquests were focused on Palestine and Greater Roman Syria. "Romanized" Arab tribes were part of these early armies - so there was already some existing cultural cross-pollination going on once the dust had settled after the conquest of Jerusalem and Damascus. One may compare it to the germanic tribes that ended up conquering the Western Roman Empire - they had (to varying degrees) Romanized elites, romanized militaries, etc.

The Roman Empire soon became the primary "enemy" of the fledging Arab Empire. Keep this important note in mind, as it would change the dynamic in how Roman and early Islamic Civilizations would interact with one another. The Arabs had conquered the Romans' most prized provinces - and within 40 years of Muhammad's death would be at the walls of Constantinople conducting their first siege of the city.

Let's move to the more obvious early impacts that Roman culture had on this nascent Arab Empire and later Islamic civilization. Hisham's Palace, The Great Mosque of Damascus, The Dome of the Rock and Al-Zaytuna Mosque are all early Arab/Islamic structures that have clear Roman architectural influence. The famous Roman arches are seen throughout Islamic architecture in North Africa and the Levant during this time period.

In terms of academic, philosophical and intellectual influence - this is another interesting realm. The influence of the Greek tradition in Islamic philosophy/theology is, in my opinion, too often romanticized by secular academics in the west. There were platonic (mostly in Shi'a Ismailism) and aristotelean (Mu'tazilites, Ibn Rushd) influences found in heterodox muslim philosophical circles, but, again I think the broader philosophical tradition of Sunni Islam, Roman/Greek influences are often exaggerated here in the West.

Regarding the Ottomans, it's true that Mehmet II considered himself the heir of Roman civilization, but this didn't trickle down into the average Ottoman muslim. Although much of the political and military elite in the Ottoman Empire would have been of Balkan European stock, this didn't change the fact that they would have identified as Muslims first and not Romans. The Greek christians would have been considered the Romans in the Ottoman empire.

Finally, The Persians. One would assume that because the Arab conquests conquered the wealthy Roman provinces of North Africa, Egypt, Syria and Palestine, that this early Islamic civilization would have had a de facto Romanized character to it. However, the Arabs would end up fully conquering an empire to the east - the vast and ancient Persian empire - that would end up influencing Islamic Civilization to a remarkable degree. Once could argue that once the Persians began converting en masse to Islam, that we start actually start to see a distinct Islamic Civilization emerge. The "Persianization" of the early Arab and eventual Islamic empires would become a phenomenon that had lasting effects through Egypt, Constantinople, Central Asia and the Indian Subcontinent. I'd argue that the Ottomans were more Persianized than Romanized. Many people don't realize that Persian, not Arabic was the lingua franca for a huge part of Islamic Civilization.

TL;DR - The early Arab Empire and later Islamic Civilization didn't really become Romanized because the Arabs would end up completely conquering and absorbing a greater empire - The Persian Empire. Islamic theology, the Arabic language and grammar, government bureaucracy, art and architecture, poetry, music, etc, were all immensely influenced by the Persians. Personally, I think if the Persians hadn't converted to Islam en masse in the 9th and 10th centuries, we wouldn't have Islamic Civilization as we know it today.

I'll end with a quote from the great North African scholar Ibn Khaldun:

"…It is a remarkable fact that, with few exceptions, most Muslim scholars…in the intellectual sciences have been non-Arabs…thus the founders of grammar were Sibawaih and after him, al-Farisi and Az-Zajjaj. All of them were of Persian descent…they invented rules of (Arabic) grammar…great jurists were Persians… only the Persians engaged in the task of preserving knowledge and writing systematic scholarly works. Thus the truth of the statement of the prophet becomes apparent, ‘If learning were suspended in the highest parts of heaven the Persians would attain it”…The intellectual sciences were also the preserve of the Persians, left alone by the Arabs, who did not cultivate them…as was the case with all crafts…This situation continued in the cities as long as the Persians and Persian countries, Iraq, Khorasan and Transoxiana (modern Central Asia), retained their sedentary culture.”