Yes, there was a process of Indian unification. That "process" was the conquest of the entire Indian subcontinent by the British East India Company, and the subsequent administration of the entirety of the subcontinent under a single, centralized government headed by the Governor-General of India (later also known as the Viceroy of India, after the British Crown assumed control of India from the East India Company).

If it weren't for the British Raj, we have little reason to believe that "India" (or rather, the non-Muslim-majority parts of India) would exist today as a unified state. For the vast majority of history, the Indian subcontinent existed as a collection of various regional kingdoms and empires; occasionally, there would be an imperial state that assumed nearly pan-Indian dimensions (like the Delhi Sultanate under Muhammad bin Tughlaq, or the Mughal Empire under Aurangzeb), but such imperial formations tended to be extremely unstable and usually dissolved after a couple decades. The British Empire represented a watershed in this respect, because it marked the first time in history that the entire subcontinent was under a stable, centralized rule for an extended period of time (long enough for effective political institutions to be established, which allowed the continuance of centralized governance even after the British left India).

Also, I would avoid using terms like "Aryan North India" or "Dravidian South India" when talking about relevant cultural or political identities in pre-modern India, or trying to understand India's lack of historical unity. Indians in pre-modern times, whether of the elite class or the lower classes, did not conceptualize themselves in such a way. These terms, in their modern usage, only became common in the 19th century as a result of work in philology and comparative linguistics, which revealed that a bunch of languages in the north of India shared a common linguistic ancestor (which was also shared with most European languages) whereas another bunch of languages in the south of India shared a completely unique and separate ancestry (what we now know as "Dravidian languages"). However, pre-modern South Indians did not identify as "Dravidians," nor did pre-modern North Indians identify as "Indo-Aryans." Their identities were based on such things as caste, region, locality, language (not language group), and religious sect.

For more information on the specific identities of Indian regions throughout history, you can see my previous post here: https://np.reddit.com/r/AskHistorians/comments/7gpg7m/why_was_india_made_a_country_while_the_middle/dql0m8h/