Are the Roman and Persian empires otherwise relatively stable?
Yes, I'd say so. The Persians are able to hold the Arabs off during the original invasion and, although they lose Mesopotamia, and have a dynastic switch, are holding themselves together and are unlikely to be conquered in the near future. Byzantium is ... well, Byzantium. They're pretty much as in OTL, although the haven't seen Anatolia overrun by the Arabs threatening Constantinople.
So, pretty much, we have a Middle East that is divided between three powers which are pretty evenly matched and stable.
And this is where my initial question comes in. So much as Arabic science and philosophy - to my understanding - came about because they ruled over such an expansive region that they were culturally influenced by so many different peoples. They had access to the Greek Classics, but also Persian and Indian math, science and philosophy. Although the Arabs (and the Persians and Byzantines, for that matter) are still going to be aware of some of these works through trade and so forth, its not like the Middle East is going to be united under a single real and all that entails.