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We present (g₄₇₅- z₈₅₀) color and z₈₅₀-band surface brightness fluctuations (SBFs) measurements for 43 early-type galaxies in the Fornax cluster imaged with the Hubble Space Telescope Advanced Camera for Surveys. These are combined with our earlier measurements for Virgo cluster galaxies to derive a revised, nonlinear calibration of the z₈₅₀-band SBF absolute magnitude as a function of (g₄₇₅- z₈₅₀) color, valid for the AB color range 0.8 < (g₄₇₅- z₈₅₀) < 1.6. In all, we tabulate recalibrated SBF distances for 134 galaxies in Virgo, Fornax, the Virgo W' group, and NGC 4697 in the Virgo Southern Extension. The calibration procedure yields a highly precise relative distance modulus for Fornax with respect to Virgo of Δ(m - M)_FV= 0.42±0.03 mag, or a distance ratio d_F/d_V = 1.214±0.017. The resulting Fornax distance modulus is (m - M)_For= 31.51±0.03±0.15 mag, corresponding to d_F= 20.0±0.3±1.4 Mpc, where the second set of error bars reflects the total systematic uncertainty from our assumed Virgo distance of 16.5 Mpc. The rms distance scatter for the early-type Fornax cluster galaxies is σ_d= 0.49^+0.11_–0.15 Mpc, or a total line-of-sight depth of 2.0^+0.4_–0.6Mpc, consistent with its compact appearance on the sky. This translates to a depth scatter smaller than the intrinsic, or "cosmic," scatter σ_cos in the SBF calibration, unlike the case for the larger Virgo cluster. As a result, we are able to place the first tight constraints on the value of σ_cos. We find σ_cos = 0.06±0.01 mag, with a firm upper limit of σ_cos< 0.08 mag, for the subsample of galaxies with (g₄₇₅- z₈₅₀)>1.02, but it is about twice as large for bluer galaxies. We also present an alternative SBF calibration based on the "fluctuation count" parameter, a proxy for galaxy mass. This gives a consistent relative distance but with larger intrinsic scatter, and we adopt the result from the calibration on (g₄₇₅- z₈₅₀) because of its basis in stellar population properties alone. Finally, we find no evidence for systematic trends of the galaxy distances with position or velocity (e.g., no current infall); the Fornax cluster appears both compact and well virialized.