  We examine how corrections to $S$-state energy levels, $ E_{nS}$, in
hydrogenic atoms due to the finite proton size are affected by moments of the
proton charge distribution. The corrections to $E_{nS}$ are computed moment by
moment. The results demonstrate that the next-to-leading order term in the
expansion is of order $r_p / a_B $ times the size of the leading order $
\langle r_p^2 \rangle $ term. Our analysis thus dispels any concern that the
larger relative size of this term for muonic hydrogen versus electronic
hydrogen might account for the current discrepancy of proton radius
measurements extracted from the two systems. Furthermore, the next-to-leading
order term in powers of $r_p / a_B $ that we derive from a dipole proton form
factor is proportional to $\langle r_p^3 \rangle $, rather than $\langle r_p^4
\rangle$ as would be expected from the scalar nature of the form factor. The
dependence of the finite-size correction on $\langle r_p^3 \rangle $ and higher
odd-power moments is shown to be a general result for any spherically symmetric
proton charge distribution. A method for computing the moment expansion of the
finite-size correction to arbitrary order is introduced and the results are
tabulated for principal quantum numbers up to $n=7$.
