  The Iron Calorimeter (ICAL) detector at the proposed India-based Neutrino
Observatory (INO) aims to detect atmospheric neutrinos and antineutrinos
separately in the multi-GeV range of energies and over a wide range of
baselines. By utilizing its charge identification capability, ICAL can
efficiently distinguish $\mu^-$ and $\mu^+$ events. Atmospheric neutrinos
passing long distances through Earth can be detected at ICAL with good
resolution in energy and direction, which enables ICAL to see the
density-dependent matter oscillations experienced by upward-going neutrinos in
the multi-GeV range of energies. In this work, we explore the possibility of
utilizing neutrino oscillations in the presence of matter to extract
information about the internal structure of Earth complementary to seismic
studies. Using good directional resolution, ICAL would be able to observe 331
$\mu^-$ and 146 $\mu^+$ core-passing events with 500 kt$\cdot$yr exposure. With
this exposure, we show for the first time that the presence of Earth's core can
be independently confirmed at ICAL with a median $\Delta \chi^2$ of 7.45 (4.83)
assuming normal (inverted) mass ordering by ruling out the simple two-layered
mantle-crust profile in theory while generating the prospective data with the
PREM profile. If we generate the data with the simple three-layered
core-mantle-crust profile of the Earth, the above-mentioned $\Delta \chi^2$
changes to 6.31 (3.92) assuming normal (inverted) mass ordering.
