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Supernovae and their remnants provide energetic feedback to the ambient interstellar medium (ISM), which is often distributed in multiple gas phases. Among them, warm molecular hydrogen (H$_2$) often dominates the cooling of the shocked molecular ISM, which has been observed with the H$_2$ emission lines at near-infrared wavelengths. Such studies, however, were either limited in narrow filter imaging or sparsely sampled mid-infrared spectroscopic observations with relatively poor angular resolutions. Here we present near-infrared ($H$- and $K$-band) spectroscopic mosaic observations towards the A, B, C, and G regions of the supernova remnant (SNR) IC 443, with the K-band Multi-Object Spectrograph (KMOS) onboard the Very Large Telescope (VLT). We detected 20 ro-vibrational transitions of H$_2$, one H line (Br$γ$), and two [Fe II] lines, which dominate broadband images at both $H$- and $K$-band. The spatial distribution of H$_2$ lines at all regions are clumpy on scales from $\sim 0.1$ pc down to $\sim 0.008$ pc. The fitted excitation temperature of H$_2$ is between 1500 K and 2500 K, indicating warm shocked gas in these regions. The multi-gas-phase comparison shows stratified shock structures in all regions, which explains the co-existence of multiple types of shocks in the same regions. Last, we verify the candidates of young stellar objects previously identified in these regions with our spectroscopic data, and find none of them are associated with young stars. This sets challenges to the previously proposed scenario of triggered star formation by SNR shocks in IC~443.