Linelike features in TeV γ rays constitute a "smoking gun" for TeV-scale particle dark matter and new physics. Probing the Galactic Center region with ground-based Cherenkov telescopes enables the search for TeV spectral features in immediate association with a dense dark matter reservoir at a sensitivity out of reach for satellite γ-ray detectors, and direct detection and collider experiments. We report on 223 hours of observations of the Galactic Center region with the MAGIC stereoscopic telescope system reaching γ-ray energies up to 100 TeV. We improved the sensitivity to spectral lines at high energies using large-zenith-angle observations and a novel background modeling method within a maximum-likelihood analysis in the energy domain. No linelike spectral feature is found in our analysis. Therefore, we constrain the cross section for dark matter annihilation into two photons to ⟨σv⟩≲5×10^{-28} cm^{3} s^{-1} at 1 TeV and ⟨σv⟩≲1×10^{-25} cm^{3} s^{-1} at 100 TeV, achieving the best limits to date for a dark matter mass above 20 TeV and a cuspy dark matter profile at the Galactic Center. Finally, we use the derived limits for both cuspy and cored dark matter profiles to constrain supersymmetric wino models.