The addition of an external starshade to the Nancy Grace Roman Space Telescope will enable the direct imaging of Earth-radius planets orbiting at ∼1 AU. Classification of any detected planets as Earth-like requires both spectroscopy to characterize their atmospheres and multi-epoch imaging to trace their orbits. We consider here the ability of the Starshade Rendezvous Probe to constrain the orbits of directly imaged Earth-like planets. The target list for this proposed mission consists of the 16 nearby stars best suited for direct imaging, around which ∼10 to 15 planets are expected to be discovered. Of these planets, ∼1 to 2 will be Earth-like in mass and temperature. The field of regard for the starshade mission is constrained by solar exclusion angles, resulting in four observing windows during a two-year mission. We find that for Earth-like planets that are detected at least three times during the four viewing opportunities, their semi-major axes are measured with a median precision of 7 mas, or a median fractional precision of 3%. Habitable-zone planets can be correctly identified as such 96.7% of the time, with a false positive rate of 2.8%. If a more conservative criteria are used for habitable-zone classification (95% probability), the false positive rate drops close to zero, but with only 81% of the truly Earth-like planets correctly classified as residing in the habitable zone.