We test Milgromian dynamics (MOND) using wide binary stars (WBs) with separations of $2-30$ kAU. Locally, the WB orbital velocity in MOND should exceed the Newtonian prediction by $\approx 20\%$ at asymptotically large separations given the Galactic external field effect (EFE). We investigate this with a detailed statistical analysis of \emph{Gaia} DR3 data on 8611 WBs within 250 pc of the Sun. Orbits are integrated in a rigorously calculated gravitational field that directly includes the EFE. We also allow line of sight contamination and undetected close binary companions to the stars in each WB. We interpolate between the Newtonian and Milgromian predictions using the parameter $\alpha_{\rm{grav}}$, with 0 indicating Newtonian gravity and 1 indicating MOND. Directly comparing the best Newtonian and Milgromian models reveals that Newtonian dynamics is preferred at $19\sigma$ confidence. Using a complementary Markov Chain Monte Carlo analysis, we find that $\alpha_{\rm{grav}} = -0.021^{+0.065}_{-0.045}$, which is fully consistent with Newtonian gravity but excludes MOND at $16\sigma$ confidence. This is in line with the similar result of Pittordis and Sutherland using a somewhat different sample selection and less thoroughly explored population model. We show that although our best-fitting model does not fully reproduce the observations, an overwhelmingly strong preference for Newtonian gravity remains in a considerable range of variations to our analysis. Adapting the MOND interpolating function to explain this result would cause tension with rotation curve constraints. We discuss the broader implications of our results in light of other works, concluding that MOND must be substantially modified on small scales to account for local WBs.