No, the word "dollar" is quite a bit older than the US government's unit of currency. It is an anglicised form of "thaler", which was the name given to coins first minted in 1519 from silver mined in Joachimsthal in Bohemia. Thaler is a shortened form of the term by which the coin was originally known: Joachimsthaler.

Eventually, "dollar" came to apply to lots of very similar coins, not only ones minted in central Europe but also the Spanish peso and the Portuguese eight-real piece. Both these large silver coins were practically identical in weight and fineness. This is the origin of phrase pieces of eight.

The word was in general use in England well before the creation of the US (actually when the English colonisation of North America had scarcely begun) and can even be found in two of Shakespeare's plays: the Tempest and Macbeth.

In Macbeth, Rosse says:

That now
Sweno, the Norway's King, craves composition;
Nor would we deign him burial of his men
Till he disbursed at Saint Colme's Inch
Ten thousand dollars to our general use.

Source: A History of Money from Ancient Times to the Present Day, Glyn Davies.