Using the method that was developed in the first paper of this series, we measured the vertical gravitational potential of the Galactic disk from the time-varying structure of the phase-space spiral, using data from Gaia as well as supplementary radial velocity information from legacy spectroscopic surveys. For eleven independent data samples, we inferred gravitational potentials that were in good agreement, despite the data samples' varied and substantial selection e ffects. Using a model for the baryonic matter densities, we inferred a local halo dark matter density of 0.0085 +/- 0.0039 M(circle dot)pc(-3) = 0.32 +/- 0.15 GeV cm(-3). We were also able to place the most stringent constraint on the surface density of a thin dark disk with a scale height