We analyze 88 independent, high-resolution, cosmological zoomed-in simulations of disk galaxies in the NIHAO simulations suite to explore the connection between the atomic gas fraction and angular momentum (AM) of baryons throughout cosmic time. The study is motivated by the analytical model of Obreschkow et al., which predicts a relation between the atomic gas fraction f(atm) and the integrated atomic stability parameter q = j sigma/(GM), where M and j are the mass and specific AM of the galaxy (stars+cold gas) and sigma is the velocity dispersion of the atomic gas. We show that the simulated galaxies follow this relation from their formation (z similar or equal to 4) to the present within similar to 0.5 dex. To explain this behavior, we explore the evolution of the local Toomre stability and find that 90%-100% of the atomic gas in all simulated galaxies is stable at any time. In other words, throughout the entire epoch of peak star formation until today, the timescale for accretion is longer than the timescale to reach equilibrium, thus resulting in a quasi-static equilibrium of atomic gas at any time. Hence, the evolution of f(atm) depends on the complex hierarchical growth history primarily via the evolution of q. An exception is galaxies subject to strong environmental effects.