Long story short :
Yes it can exist without fiat but not right now.
~
It's a seperate entity and you might live in a country where bitcoins are used very often and you are able to pay with it in every corner but in most countries, it's not the option, bitcoins is used but the only way you can use it is by exchanging or trading online or offline.
Yes, right now value of BTC is measured in fiat. So its difficult for BTC to exist on its own right. Going back to Satoshi whitepaper, bitcoin was designed as p2p cash transfer system. BTC has not token its own value yet, like we cant say this car worth is 1.5 BTC or a cup of coffee cost 0.0001 BTC.
That's just my 2 cents