We report a search for nonstandard neutrino interactions (NSI) using eight years of TeV-scale atmospheric muon neutrino data from the IceCube Neutrino Observatory. By reconstructing incident energies and zenith angles for atmospheric neutrino events, this analysis presents unified confidence intervals for the NSI parameter ε_{μτ}. The best-fit value is consistent with no NSI at a p value of 25.2%. With a 90% confidence interval of -0.0041≤ε_{μτ}≤0.0031 along the real axis and similar strength in the complex plane, this result is the strongest constraint on any NSI parameter from any oscillation channel to date.