PPP is usually meant to compare economies and control for differences in cost of living, not comparing time periods. Inflation is when you compare the same economy in two time periods.

Furthermore, economic growth is measured in "real" dollars, aka, excludig inflation. If US prices doubled as you say, then their economy is therefore now four times as large in the new, inflated dollars.

None of this has anything to do with fiat currency. The same would be true if you measured in kg's of gold, but you'd have to account for the gold-cost of living ratio instead of fiat dollar inflation.