  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$\gamma$), 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.
