Paleodetectors are a proposed experimental technique for direct detection of dark matter (DM) via the readout of DM-induced nuclear recoil tracks in natural minerals. The large detector mass required for the sensitivity of conventional direct detection experiments to rare events is replaced by the exposure of paleodetectors to DM-induced nuclear recoils over geological timescales. In this paper, we generalize the previous theoretical predictions that focused on canonical spin-independent coherent and spin-dependent scattering (proportional to A2 and the spin of the nucleus, respectively). We estimate the sensitivity of paleodetectors to interactions between weakly interacting massive particle (WIMP) DM and nuclei within the framework of a nonrelativistic effective field theory (NREFT), considering isoscalar couplings to nucleons for both elastic and inelastic scattering. Taking into account cosmogenic, astrophysical, and radiogenic backgrounds, we project the 90% confidence-level upper limits on the isoscalar NREFT coupling constants for both scattering types. We consider representative readout scenarios and examine a wide variety of target minerals. The projected sensitivities of paleodetectors are compared with the 90% confidence-level limits from the XENON100, LUX-ZEPLIN, and PandaX-II experiments, as well as with the 95% Bayesian credible region of the two-dimensional marginalized posterior distribution from SuperCDMS. For DM masses from 1 GeV/c2 - 10 GeV/c2, we find that paleodetectors are projected to have sensitivity superior to that of conventional experiments for WIMP-nucleus interactions mediated by all NREFT operators, largely independent of readout scenario or target mineral. For heavier DM masses in the range 10 GeV/c2 - 5 TeV/c2, we find that the sensitivity of paleodetectors is projected to be comparable to or better than that of conventional experiments for WIMP-nucleus interactions mediated by several NREFT operators, depending on the readout scenario and target mineral.