The Sun provides an excellent target for studying spin-dependent dark matter-proton scattering due to its high matter density and abundant hydrogen content. Dark matter particles from the Galactic halo can elastically interact with Solar nuclei, resulting in their capture and thermalization in the Sun. The captured dark matter can annihilate into Standard Model particles including an observable flux of neutrinos. We present the results of a search for low-energy ($<$ 500 GeV) neutrinos correlated with the direction of the Sun using 7 years of IceCube data. This work utilizes, for the first time, new optimized cuts to extend IceCube's sensitivity to dark matter mass down to 5 GeV. We find no significant detection of neutrinos from the Sun. Our observations exclude capture by spin-dependent dark matter-proton scattering with cross-section down to a few times $10^{-41}$ cm$^2$, assuming there is equilibrium with annihilation into neutrinos/anti-neutrinos for dark matter masses between 5 GeV and 100 GeV. These are the strongest constraints at GeV energies for dark matter annihilation directly to neutrinos.