We have observed the spectrum of Procyon A (F5 IV) from 4559 to 5780 Å with a signal-to-noise ratio of ~103 and a resolving power of 2×105. We have measured the line bisectors and relative line shifts of a large number of Fe I and Fe II lines, comparing them to those found in the solar spectrum. A three-dimensional hydrodynamical model atmosphere has been computed and is tested against observations. The model reproduces in detail most of the features observed, although we identify some room for improvement. At all levels, the comparison of the three-dimensional time-dependent calculations with the observed spectral lines shows a much better agreement than for classical homogeneous models, making it possible to refine previous estimates of the iron abundance, the projected rotational velocity, the limb darkening, and the systemic velocity of the Procyon binary system. The difference between the iron abundance determined with the three-dimensional model and its one-dimensional counterpart is <~0.05 dex. We find consistency between the iron abundance derived from Fe I and Fe II lines, suggesting that departures from LTE in the formation of the studied lines are relatively small. The scatter in the iron abundance determined from different lines still exceeds the expectations from the uncertainties in the atomic data, pointing out that one or more components in the modeling can be refined further.