We detail the sensitivity of the liquid xenon (LXe) DARWIN observatory to solar neutrinos via elastic electron scattering. We find that DARWIN will have the potential to measure the fluxes of five solar neutrino components: $pp$, $^7$Be, $^{13}$N, $^{15}$O and $pep$. The precision of the $^{13}$N, $^{15}$O and $pep$ components is hindered by the double-beta decay of $^{136}$Xe and, thus, would benefit from a depleted target. A high-statistics observation of $pp$ neutrinos would allow us to infer the values of the weak mixing angle, $sin^2theta_w$, and the electron-type neutrino survival probability, $P_e$, in the electron recoil energy region from a few keV up to 200 keV for the first time, with relative precision of 5% and 4%, respectively, at an exposure of 300 ty. An observation of $pp$ and $^7$Be neutrinos would constrain the neutrino-inferred solar luminosity down to 0.2%. A combination of all flux measurements would distinguish between the high (GS98) and low metallicity (AGS09) solar models with 2.1-2.5$sigma$ significance, independent of external measurements from other experiments or a measurement of $^8$B neutrinos through coherent elastic neutrino-nucleus scattering in DARWIN. Finally, we demonstrate that with a depleted target DARWIN may be sensitive to the neutrino capture process of $^{131}$Xe.
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