Depends on what extent you mean by the very term "Eastern"; even though the eastern half of Eurasia is vast and in many ways more diverse than the Western half (e.g. Islam and Christianity are more similar than Hinduism and Confucianism, for example), it seems that you were .
Also, do you mean adopting to a greater or lesser extent much of the main parts of an entire cultural package of "mother civilization" - as Japan, Korea and Vietnam did with the cultural package of the Chinese, or the rest of Southeast Asia (besides Vietnam) did with India? Or do you mean just selectively adopting certain aspects of culture, government, economy, technology or what have you? Both seem feasible for a rather wide swath of time, given the West's or the Middle East's less advanced nature when compared with India or China, from the Fall of Rome to the Enlightenment or the Industrial Revolution even for the former or several periods, the last (before Western Industrialization) being after the sack of Baghdad.
Given your original post, and the fact that I know more about Chinese history than India for the most part, I will stick with China.
The most obvious is the Song having an industrial revolution before the Mongols can occupy China; if this path is taken, it becomes more a question of how long it would take China to inherit the Earth rather than a question of cultural diffusion, given this ATL China's incredible lead on everyone else and the fact that, unlike the OTL West's New Imperialism, they are completely unified under one strong, central government that would by this point easily overshadow the British Empire at its height in all major areas of civilization and development.
Another possibility is to have the Ming Voyages of Zheng He carry on (perhaps someone assassinates the Hongxi Emperor before he can be enthroned? Many possibilities exist here.) At this point, the West, though seeing the beginnings of the Renaissance in the more advanced states in the south, simply could stop China in the long run, even if they unified to fight the "heathens" (nearly impossible, given the lack of solid history of centralized government, in contrast to China), there would simply be no way around the Chinese, nor any way of challenging spectacularly superior Chinese ships in a large-scale fight.
Looking back to the Tang Dynasty, if there is no Islam, perhaps due to an Axumite offensive, the Chinese could expand their sphere of influence into Central Eurasia and maybe even the Sassanids, given their obviously quite feeble state around the time of the Islamic Conquest of Persia: If the Arabs could conquer the Persians, than certainly the Tang could at least force it into tributary state status.
