We present a rapid analytic framework for predicting kilonova light curves following neutron star (NS) mergers, where the main input parameters are binary-based properties measurable by gravitational wave detectors (chirp mass and mass ratio, orbital inclination) and properties dependent on the nuclear equation of state (tidal deformability, maximum NS mass). This enables synthesis of a kilonova sample for any NS source population, or determination of the observing depth needed to detect a live kilonova given gravitational wave source parameters in low latency. We validate this code, implemented in the public MOSFiT package, by fitting it to GW170817. A Bayes factor analysis overwhelmingly ($B>10^{10}$) favours the inclusion of an additional luminosity source in addition to lanthanide-poor dynamical ejecta during the first day. This is well fit by a shock-heated cocoon model, though differences in the ejecta structure, opacity or nuclear heating rate cannot be ruled out as alternatives. The emission thereafter is dominated by a lanthanide-rich viscous wind. We find the mass ratio of the binary is $q=0.92pm0.07$ (90% credible interval). We place tight constraints on the maximum stable NS mass, $M_{rm TOV}=2.17^{+0.08}_{-0.11}$ M$_odot$. For a uniform prior in tidal deformability, the radius of a 1.4 M$_odot$ NS is $R_{1.4}sim 10.7$ km. Re-weighting with a prior based on equations of state that support our credible range in $M_{rm TOV}$, we derive a final measurement $R_{1.4}=11.06^{+1.01}_{-0.98}$ km. Applying our code to the second gravitationally-detected neutron star merger, GW190425, we estimate that an associated kilonova would have been fainter (by $sim0.7$ mag at one day post-merger) and declined faster than GW170817, underlining the importance of tuning follow-up strategies individually for each GW-detected NS merger.
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