For that matter, India was rarely unified. I don't think a civ has to be politically unified, but it should be culturally unified.
India is not culturally unified,I think that statement is not supported by evidence,history. It can be said political unity is rarer wrt. to China or Iran.
The guy who came up with civilizational state concept for China Mr.Zhang Wei Wei also don't classied India as single civilization though Huntington put it as one civilization,so may be at the end what matters is one perception or idea of term civilization.
In case of India,there is a lot of cultural unity and it becomes more clearer when u see Hinduism more than just a religion. It is present in every dimension of life whether culture,science,polity,society structure. A very important unifying factor is Sanskrit and it's traditions,literature,philosophy etc. Though it was not a mass language at least for regions far from core Vedic regions,it was a connecting language which ensured transfer of culture all over India.
Examples can be seen through out history whether it is presence of Vedic deities,Idealisation of Ram Rajya(Rule)(chola texts) in Tamil region or propagation of Bhakti Movement from Tamil Nadu to rest of India which shaped North India during Muslim rule.
The Concept of sacred geography also play a very important role as it gives a geographical sense to masses,whether it is sacred Ganga or Pilgrimage to temple,mountains etc .One can understand its value by the fact when Maratha finally manage to break Mughal power they renovated city of Varanasi on bank of Ganga and brought Marathi Brahmins to settle down there. Chola king brought North Indian brahmins with him for his campaign in North.
In terms of Polity,the concept Dharma in accordance to rta(cosmic order) is underlying theme. King is not seen as having divine power but as protector of rta in accordance to Dharma guided by Dharmashasta,Arthasashtra. One can easily see similarity in their conducts like patronising all religions Hinduism,Buddhism,Jainism irrespective of King's own religion. Many dynasties all over India claimed they r part of kshatriya traditions to gain legitimacy. Asvamedha is a good example of continuity from Vedic period to last happed in 1741 by rajput King.
In terms of Sceince one can see continuity from Vedic time to last school of Indic Science in Kerala. Indian kings patronized science and India was among leader before invasions especially in field of Mathematics,Botany and Medicines.
So there is enough continuity,unity in terms of culture. One can easily see the impact in today's time there r region which r much closer to Delhi but separated from India. Region much closer to capital and there is insurgency while region comparatively far away in south but not any such separatism on the other hand right now most contributing region in India's growth.