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Freshly-synthesized r-process elements in kilonovae ejecta imprint absorption features on optical spectra, as observed in the GW170817 binary neutron star merger. These spectral features encode insights into the physical conditions of the r-process and the origins of the ejected material, but associating features with particular elements and inferring the resultant abundance pattern is computationally challenging. We introduce Spectroscopic r-Process Abundance Retrieval for Kilonovae (SPARK), a modular framework to perform Bayesian inference on kilonova spectra with the goals of inferring elemental abundance patterns and identifying absorption features at early times. SPARK inputs an atomic line list and abundance patterns from reaction network calculations into the TARDIS radiative transfer code. It then performs fast Bayesian inference on observed kilonova spectra by training a Gaussian process surrogate for the approximate posteriors of kilonova ejecta parameters, via active learning. We use the spectrum of GW170817 at 1.4 days to perform the first inference on a kilonova spectrum, and recover a complete abundance pattern. Our inference shows that this ejecta was generated by an r-process with either (1) high electron fraction Y_e ~ 0.35 and high entropy s/k_B ~ 25, or, (2) a more moderate Y_e ~ 0.30 and s/k_B ~ 14. These parameters are consistent with a shocked, polar dynamical component, and a viscously-driven outflow from a remnant accretion disk, respectively. We also recover previous identifications of strontium absorption at ~8000 AA, and tentatively identify yttrium and/or zirconium at < 4500 AA. Our approach will enable computationally-tractable inference on the spectra of future kilonovae discovered through multi-messenger observations.