Assume that the Indo-Greek Kingdom survives, either by having Eucratides not be assassinated (so he can complete his conquest of his rival indo-greek kingdom, ruled by Demetrius) or by giving a stabler succession to Menander/Milinda.
What are the effects on:
The connections between Europe, the Near East, and India?
The spread of buddhism?
As
@LSCatilina notes, it wasn't altogether a united polity. I believe Menander had to keep himself from conquering the Shunga Empire in Maghada because he was attacked by a rival Bactrian king. Now, Menander got pretty close to doing conquering Maghada, even raiding Pataliputra, which is quite far down the Ganges. So, conquest is quite possible. However, then we get into the whole affair of the Parthians, and so we really need the Indo-Greeks to conquer some ports for further trade and connections with Europe.
Speaking of which, if they do so, it's quite possible we get trade links as strong as they'd be with the later Romans, with Romans complaining about India taking gold as Pliny did so IOTL, quite a bit earlier than OTL. Buddhism would likely be strengthened, and come along the trade routes, but it may only become a minor, insignificant religion in the diverse Roman state, which may have been OTL as various excavations in Alexandria have actually revealed statues of the Buddha. However, if it is able to collide with the preexisting surge of Neopythagoreanism (which it was actually quite similar), then Buddhism could become a major phenomenon and school of thought - except, Buddhist missionaries would try to convert the poor with their acts of charity, and this brings us to a possibility of Buddhism spreading as a less disruptive form of Christianity. However, this is not the most plausible event at all in this scenario.
What is plausible is more fusion between East and West in India. I could easily see the Indo-European family discovered two thousand years earlier with this scenario. However, even IOTL, there was quite a bit of Western knowledge in India - there is a number of Greek loanwords in Sanskrit, the Indian zodiac is clearly taken from the Hellenic one, the Greek attitude towards barbarians was standard Indian thought until the days of the Delhi Sultanate, the Ptolemaic model makes an appearance in the
Aryabhatiya, etc.
I'm not sure if this massive impact could even get any larger. Except for one thing. India's written tradition was not the strongest at this point. The simpler Greek script displacing the Brahmi script is possible, despite all its flaws. Indeed, the Greek script was used by the Bactrians right up until the arrival of Islam even though it failed to represent all of the Bactrian sounds. If that's what happens in India, my oh my, now that's a massive butterfly.
Will there be further greek migration into the kingdom? Or must the aristocracy assimilate into the more native populace to survive?
Well, the answer is that it will assimilate. India has done that to invaders, from the Hepthalites to the Turks. But, in all circumstances, those invaders were absorbed, though they all brought elements of their own cultures into India. Just look at the strength of Persianate culture in India. Now, of course Greek culture will absorb much better than Persianate culture, but it will leave a larger imprint on Indian society - and maybe a more obvious one, as well.
