Low-frequency gravitational-wave experiments such as the Laser Interferometer Space Antenna and pulsar timing arrays are expected to detect individual massive black hole (MBH) binaries and mergers. However, secure methods of identifying the exact host galaxy of each MBH merger among the large number of galaxies in the gravitational-wave localization region are currently lacking. We investigate the distinct morphological signatures of MBH merger host galaxies, using the Romulus25 cosmological simulation. We produce mock telescope images of 201 simulated galaxies in Romulus25 hosting recent MBH mergers through stellar population synthesis and dust radiative transfer. Based on comparisons to mass- and redshift-matched control samples, we show that combining multiple morphological statistics via a linear discriminant analysis enables identification of the host galaxies of MBH mergers, with accuracies that increase with chirp mass and mass ratio. For mergers with high chirp masses (≳108.2M⊙) and high mass ratios (≳0.5), the accuracy of this approach reaches ≳80%, and does not decline for at least ∼1 Gyr after numerical merger. We argue that these trends arise because the most distinctive morphological characteristics of MBH merger and binary host galaxies are prominent classical bulges, rather than relatively short-lived morphological disturbances from their preceding galaxy mergers. Since these bulges are formed though major mergers of massive galaxies, they lead to (and become permanent signposts for) MBH binaries and mergers that have high chirp masses and mass ratios. Our results suggest that galaxy morphology can aid in identifying the host galaxies of future MBH binaries and mergers.