If a fluctuating cryptocurrency becomes the new world coin, would it stop fluctuating? An often mentioned criterium for money is that it should be a unit of account. It could be argued that cryptocurrencies don't satisfy this requirement.

The argument is that businesses like bakers would adjust their prices all the time, which is not acceptable. I.e. it fails to be a unit of account.

Suppose that suddenly a cryptocurrency would be the only coin that's accepted in this world.

Would this mean that it stops fluctuating, as there's nothing else to exchange the coin into, and thus becomes a unit of account?
Your question is very similar to "if we were back on the gold standard, what mechanisms exist to regulate inflation/deflation".

The question is a difficult one because it brings into scope the role of banks and credit and how they would fit into the world. Crypto-systems (in the sense of cryptocurrencies) don't address credit today.

You'd likely see the world separated into two categories of 'user':


Those who manage their own wallets
Those who do not


For the second category banks would continue to exist. Cold wallet storage, data centers and the like. Those banks would still be licensed to issue notes on the basis of the 'digital gold' they hold, so we'd be back to square one with banks being literally issuers of notes with cryptocoin as the new gold.

The value of the trustless gold, and its trust based proxy, would be regulated by the same mechanisms that existed under the old gold framework: trade imbalances result in cross border settlements of gold influx/egress that adjust bank exposure to circulating 'paper'. This results in credit interest rate adjustments, which accelerate or brake the economy depending.

So yes in short international trade would act as a mechanism for adjusting things and tending towards some regulation of the currency value. Similarly central banks would adopt policies to manage the availability of coins. I don't know what policies, but we should never underestimate the power of states (eg now under C19) to simple dictate that coins be donated/given/taxed etc to a central authority.