Recent theoretical studies argue that the rate of stochastic ion heating in
low-frequency Alfv\'en-wave turbulence is given by $Q_\perp = c_1 [(\delta u)^3
/\rho] \exp(-c_2/\epsilon)$, where $\delta u$ is the rms turbulent velocity at
the scale of the ion gyroradius $\rho$, $\epsilon = \delta u/v_{\perp \rm i}$,
$v_{\perp \rm i}$ is the perpendicular ion thermal speed, and $c_1$ and $c_2$
are dimensionless constants. We test this theoretical result by numerically
simulating test particles interacting with strong reduced magnetohydrodynamic
(RMHD) turbulence. The heating rates in our simulations are well fit by this
formula. The best-fit values of $c_1$ are $\sim 1$. The best-fit values of
$c_2$ decrease (i.e., stochastic heating becomes more effective) as the grid
size and Reynolds number of the RMHD simulations increase. As an example, in a
$1024^2 \times 256$ RMHD simulation with a dissipation wavenumber of order the
inverse ion gyroradius, we find $c_2 = 0.21$. We show that stochastic heating
is significantly stronger in strong RMHD turbulence than in a field of randomly
phased Alfv\'en waves with the same power spectrum, because coherent structures
in strong RMHD turbulence increase orbit stochasticity in the regions where
ions are heated most strongly. We find that $c_1$ increases by a factor of
$\sim 3$ while $c_2$ changes very little as the ion thermal speed increases
from values $\ll v_{\rm A}$ to values $\sim v_{\rm A}$, where $v_{\rm A}$ is
the Alfv\'en speed. We discuss the importance of these results for
perpendicular ion heating in the solar wind.