Supernovae from core collapse of massive stars drive shocks into the molecular clouds from which the stars formed. Such shocks affect future star formation from the molecular clouds, and the fast-moving, dense gas with compressed magnetic fields is associated with enhanced cosmic rays. This paper presents new theoretical modeling, using the Paris–Durham shock model, and new observations at high spectral resolution, using the Stratospheric Observatory for Infrared Astronomy, of the H2 S(5) pure rotational line from molecular shocks in the supernova remnant IC 443. We generate MHD models for nonsteady-state shocks driven by the pressure of the IC 443 blast wave into gas of densities $10^{3}-10^{5}\,\rm cm^{-3}$. We present the first detailed derivation of the shape of the velocity profile for emission from H 2 lines behind such shocks, taking into account the shock age, preshock density, and magnetic field. For preshock densities $10^{3}-10^{5}\,\rm cm^{-3}$, the H2 emission arises from layers that extend 0.01–0.0003 pc behind the shock, respectively. The predicted shifts of line centers, and the line widths, of the H 2 lines range from 20–2 and 30–4 km s−1 , respectively. The a priori models are compared to the observed line profiles, showing that clumps C and G can be explained by shocks into gas with density $10^{3}$ to $2\times 10^{4}\,\rm cm^{−3}$ and strong magnetic fields. Two positions in clump B were observed. For clump B2 (a fainter region near clump B), the H2 spectrum requires a J-type shock into moderate-density gas (∼$10^{2}\,\rm cm^{−3}$ ) with the gas accelerated to 100 km s−1 from its preshock location. Clump B1 requires both a magnetic-dominated C-type shock (like for clumps C and G) and a J-type shock (like for clump B2) to explain the highest observed velocities. The J-type shocks that produce high-velocity molecules may be locations where the magnetic field is nearly parallel to the shock velocity, which makes it impossible for a C-type shock (with ions and neutrals separated) to form.