Aims: We study the modification of the classical criterion for the linear onset and growth rate of the Rayleigh-Taylor instability (RTI) in a partially ionized (PI) plasma in the one-fluid description by considering a generalized induction equation. Methods: The governing linear equations and appropriate boundary conditions, including gravitational terms, are derived and applied to the case of the RTI in a single interface between two partially ionized plasmas. The boundary conditions lead to an equation for the frequencies in which some have positive complex parts, marking the appearance of the RTI. We study the ambipolar term alone first, extending the result to the full induction equation later. Results: The configuration is always unstable because of the presence of a neutral species. In the classical stability regime, the growth rate is small, since the collisions prevent the neutral fluid to fully develop the RTI. For parameters in the classical instability regime, the growth rate is lowered, but the differences with the compressible MHD case are small for the considered theoretical values of the collision frequencies and diffusion coefficients for solar prominences. Conclusions: The PI modifies some aspects of the linear RTI instability, since it takes into account that neutrals do not feel the stabilizing effect of the magnetic field. For the set of parameters representative for solar prominences, our model gives the resulting timescale comparable to observed lifetimes of RTI plumes.