Bitcoin exists without fiat, but it is a big question whether it would've reached to the present level without the pairing of fiat. People aren't aware about bitcoin and it's good. This means we can't expect bitcoin to grow. At the beginning everyone will find it difficult to believe in the bitcoin for different reasons. So, it is quite hard to have it on direct usage. Now people who have used to it are aware of its growth. Then onwards Bitcoin can exist without fiat, but the growth won't be effective.
The problem is not with the growth, but with the potential use of Bitcoin for making purchases. Imagine it, if there was no fiat, but the price changed, a person would have had to pay double of the regular price for the same item due to volatility because there was nothing to fix the price onto.
But I think there’s no need for fiat to solve that problem. Simply salesman would have to set the internal price of goods/services and make it automatically change whenever the price of Bitcoin would change. It’s very easy.