There is a very simple reason here: they're not allowed to, and it's not specific to India. From Wikipedia:
Of the diplomatic missions accredited to the Holy See, 89 are situated in Rome, although those countries, if they also have an embassy to Italy, then have two embassies in the same city, since, by agreement between the Holy See and Italy, the same person cannot at the same time be accredited to both. The United Kingdom recently housed its embassy to the Holy See in the same building as its embassy to the Italian Republic, a move that led to a diplomatic protest from the Holy See. An ambassador accredited to a country other than Italy can be accredited also to the Holy See. For example, the embassy of India, located in Bern, to Switzerland and Liechtenstein is also accredited to the Holy See while the Holy See maintains an Apostolic Nunciature in New Delhi.
Wikipedia does not cite this statement, but this article confirms it and gives the origin as the Lateran Treaty, plus a general reasoning for it:
The ban on one person doubling as ambassador to Italy and the Holy See dates to the Lateran Treaty, and it’s designed to guarantee the independence and sovereignty of the Holy See.
I suspect the idea there was that without such a prohibition, most places would just treat the Vatican as an afterthought to their relationship with Italy.
The list gives "embassies in Rome", but these will be specifically embassies to the Holy See which are located in Rome - you can't physically fit any embassies within the Vatican - rather than the embassy to Italy doing double service.