The main ingredients of ice cream are milk, cream, sugar and eggs. A plain vanilla flavour has about 200 kilocalories (kcal) per 100g, so you would need to eat about a kilo of it per day to get enough calories. This would only give you about 30g of protein per day, which is a little low, but milk and eggs do contain all the essential amino acids your body needs.
Eggs and milk also contain vitamins A and D, and most of the B vitamins, and you could take care of the rest with different ice cream flavours. Premium brands that include real fruit puree and nuts, instead of just flavour extracts, should give you enough vitamin C and E.
The downside is that an ice cream-only diet would give you way too much saturated fat and sugar, increasing your risk of coronary heart disease and diabetes. So while you may be able to survive, it’s not a recommended dietary strategy!