We perform electronic-structure calculations based on the first-principles many-body-theory approach in order to study quasiparticle band gaps and optical absorption spectra of hydrogen-passivated zigzag SiC nanoribbons. Self-energy corrections are included using the GW approximation, and excitonic effects are included using the Bethe-Salpeter equation. We systematically study nanoribbons that have widths between 0.6 and 2.2 nm. Quasiparticle corrections widen the Kohn-Sham band gaps because of enhanced interaction effects, caused by reduced dimensionality. Zigzag SiC nanoribbons with widths larger than 1 nm exhibit half-metallicity at the mean-field level. The self-energy corrections increase band gaps substantially, thereby transforming the half-metallic zigzag SiC nanoribbons to narrow gap spin-polarized semiconductors. Optical absorption spectra of these nanoribbons get dramatically modified upon inclusion of electron-hole interactions, and the narrowest ribbon exhibits strongly bound excitons, with binding energy of 2.1 eV. Thus, the narrowest zigzag SiC nanoribbon has the potential to be used in optoelectronic devices operating in the IR region of the spectrum, while the broader ones, exhibiting spin polarization, can be utilized in spintronic applications.