Lets look at where the word India comes from:
India's name
Sindhu (sangskrit, river still exist) -> Sindh (region still exist) -> Hind (persian) -> Indikos (greek) -> Indicus (latin) -> India (English)
So "India" is an English word with a diverse origin and it is what the British used to identify this region. Question is why an independent country would follow their colonial master and their choice of word to choose the name of their country? I would not do it, I would simply drop India, an English word and choose Bharat instead, a completely indigenous word, to create a clean break with the British colonial past or even the period of Muslim rule (Hindustan), to give people a sense of pride.
There is still time for you to do it, better late than never. It is time Bharatiya's should drop the mental slavery of the white man, and it can start with the dropping of the name India for your country.