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Galaxy formation and evolution involves a variety of effectively stochastic processes that operate over different timescales. The Extended Regulator model provides an analytic framework for the resulting variability (or `burstiness') in galaxy-wide star formation due to these processes. It does this by relating the variability in Fourier space to the effective timescales of stochastic gas inflow, equilibrium, and dynamical processes influencing GMC creation and destruction using the power spectral density (PSD) formalism. We use the connection between the PSD and auto-covariance function (ACF) for general stochastic processes to reformulate this model as an auto-covariance function, which we use to model variability in galaxy star formation histories (SFHs) using physically-motivated Gaussian Processes in log SFR space. Using stellar population synthesis models, we then explore how changes in model stochasticity can affect spectral signatures across galaxy populations with properties similar to the Milky Way and present-day dwarfs as well as at higher redshifts. We find that, even at fixed scatter, perturbations to the stochasticity model (changing timescales vs overall variability) leave unique spectral signatures across both idealized and more realistic galaxy populations. Distributions of spectral features including H$α$ and UV-based SFR indicators, H$δ$ and Ca-H,K absorption line strengths, D$_n$(4000) and broadband colors provide testable predictions for galaxy populations from present and upcoming surveys with Hubble, Webb \& Roman. The Gaussian process SFH framework provides a fast, flexible implementation of physical covariance models for the next generation of SED modeling tools. Code to reproduce our results can be found at https://github.com/kartheikiyer/GP-SFH