This is a nice myth but it is demonstrably false. The $ symbol was being used for Spanish silver dollars even before the American Revolution. A major book on the history of mathematical symbols devoted a chapter to it. The $ might have been inspired by the actual design on the Spanish dollar, which had a capital S between two Roman pillars (that resembled the capital I, so the whole image looked a bit like ISI); other suggestions are that it served as a sort of monogram or abbreviation for "solidis" or something similar. In any case, it predates the first use of "United States".
Another myth is that a $ sign with one vertical stroke means something different from a $ with two vertical strokes. This is complete nonsense and unsupported by any book on banking or accounting as far back as the dollar sign existed. Type foundries (and typewriters) only offer one $ sign with any typeface; some have two strokes, some have one, it seems to be a matter of whim or aesthetics.