Yes it does. You have to pay your taxes in U.S. dollars, so people have to use it. That's one of the purposes of taxes: so that the government-issued currency is used.
Of course, banks know this (it is the core of their business) and will therefore denominate loans in USD to avoid the risk of an unfavorable conversion during such a proceeding. Such an unfavorable conversion is exactly what the creditors in the MtGox bankruptcy are facing, because the proceeding involved converting their BTC accounts into JPY according to the exchange rate in 2014. Due to the change in BTC prices recently, that means that under the law the creditors will be fully repaid using what BTC assets MtGox still has, despite the fact that the creditors would receive a far greater payout if they received the amount of BTC that had been in their accounts at the time of the hack.
Beyond debts, there are also torts. Suppose I crash my car into your house and you sue me for the damage I caused. The courts will order me to pay in USD; so again, the courts are creating USD demand by creating an obligation to pay with USD.
So the demand for USD is only indirectly due to demand for goods and services; after all, there must be some reason why merchants are pricing their wares in USD. The direct driver of demand is the law itself, hence "fiat" currency.
If you want a challenge, try explaining why the Somali dollar still held its value despite being a fiat currency that lacked a government to back up its value for well over a decade.
