We investigate properties of the interstellar medium (ISM) interacting with shocks around the Galactic supernova remnant IC443, using the results of near-infrared [FeII] and H2 line mapping with the IRSF/SIRIUS. In the present study, we newly performed H2 1-0 S(1) and 2-1 S(1) line mapping with the narrow-band filters tuned for these lines, covering the entire remnant (30x35). Combined with an [FeII] line map, our result shows that the H2 line emission is significantly detected in the southern region, while the [FeII] line emission is detected all over the remnant, suggesting that slow and fast shocks propagate through the southern region and the entire remnant, respectively. In particular, the H2 line emission is relatively strong compared to the [FeII] line emission in the southwestern region, where TeV gamma-ray emission is detected. As the strong H2 line emission indicates the dominance of the dense ISM, this result supports the scenario that the gamma-ray emission is likely to be produced through a heavy interaction between cosmic-ray protons and the dense ISM in the southwestern shell. We also find that the H2 and [FeII] line emissions show an anti-correlated spatial distribution in the same region, suggesting the presence of the clumpy ISM. Such a clumpy morphology of the ISM around IC443 may assist cosmic-ray protons to efficiently interact with large amounts of the ISM protons.