What if a strong but satisfied Indian empire feels threatened by a Mesopotamian power, say one developing a strong navy and aggressively promoting its commercial interests. And instead of conquering Persia, the Indian power makes allies of the Persians, in which over time the Indian component is dominant--succeeding by marriage into the high kingship, or perhaps if the struggle with the Mesopotamian power is drawn out enough, it becomes clear the Indians are overwhelmingly the stronger partner. So the Persians, being threatened, take refuge as a privileged superprovince of the Indian power. Then finally the integrated coalition succeeds and breaks the Mesopotamian threat, and incorporates Mesopotamia as well.
From there, moving west along the Fertile Crescent into Syria and down the Levant coast should not be too difficult. Do we want it so grandiose it dominates Anatolia, Egypt and Arabia as well? The Persian acquisition should provide goodly numbers of desert-accustomed troops; with Egypt it is largely a matter of a slugfest against mercenary armies (assuming all this is long after the great Pharaonic empires) and then a suitably astute state should not have too much trouble taking possession. Anatolia is sort of out of the way and a bit tough, unless the empire has decided to seize control of the straits to monopolize Black Sea trade I don't see why this empire would push in that direction. To be sure, the Caucasian front of the Persian element of the empire might lead to some quarrels that might seem amenable to settlement by coming round the back, pointing to a Black Sea hegemony. Arabia is probably the hardest thing to acquire and hold, just because it is so sparse.
If the original motive of the Indian Empire to acquire Mesopotamia was to suppress trade rivalry in the Indian Ocean, I don't suppose they'd be overly friendly with the Arabs either. Unless perhaps the "thalassocratic" element of the Indian Empire were in fact coastal people in relationships with the Arabs; say the Empire was originally not sea-oriented at all, but in the course of absorbing coastal assets that had a prior relationship with the Arabs, becomes the patron of the Hindu-Arab trade. In that case, the Empire might have an in into Arabia on Arab terms, just as I suggested having one into Persia on Persian terms.
Such a trajectory requires several phases of growth, it isn't created overnight by one super-conqueror. This implies the Indian core power has a sophisticated tradition of government that maintains cohesion over both vast space and a fairly long period of time, consolidating the gains of one wave of advance to secure the next wave.
Would such a power stop at the western borders of what we might call "the Middle East?" Or would it keep going? Would its traders not take it down the African coast, and east into Indonesia and beyond seeking the China trade? In the northwest, having gained the eastern shore of the Med, would that be a suitable stopping place, letting the trade of the Med come to it--or would moves against rivals on the Black Sea (that threaten the Persian portions landward) lead to a drive into southeastern Europe generally, and onto the steppes to fill out the flank of the long-standing Persian front in the north?
All history of OTl suggests it has to reach some sort of limit!