We use JWST/MIRI MRS spectroscopy of a sample of six local obscured type 1.9/2 active galactic nuclei (AGN) to compare their nuclear mid-IR absorption bands with the level of nuclear obscuration traced by X-rays. This study is the first to use sub-arcsecond angular resolution data of local obscured AGN to investigate the nuclear mid-IR absorption bands with a wide wavelength coverage (4.9-28.1 micron). All the nuclei show the 9.7 micron silicate band in absorption. We compare the strength of the 9.7 and 18 micron silicate features with torus model predictions. The observed silicate features are generally well explained by clumpy and smooth torus models. We report the detection of the 6 $\mu$m dirty water ice band (i.e., a mix of water and other molecules such as CO and CO$_2$) at sub-arcsecond scales (~0.26 arcsec at 6 micron; inner ~50 pc) in a sample of local AGN with different levels of nuclear obscuration in the range log N_H (cm^-2)~22-25. We find a good correlation between the 6 micron water ice optical depths and N_H. This result indicates that the water ice absorption might be a reliable tracer of the nuclear intrinsic obscuration in AGN. The weak water ice absorption in less obscured AGN (log N_H (cm^-2)~<23.0 cm^-2) might be related to the hotter dust temperature (>T~110 K) expected to be reached in the outer layers of the torus due to their more inhomogeneous medium. Our results suggest it might be necessary to include the molecular content, such as, H$_2$O, aliphatic hydrocarbons (CH-) and more complex PAH molecules in torus models to better constrain key parameters such as the torus covering factor (i.e. nuclear obscuration).