The proposed 50 kt Iron Calorimeter (ICAL) detector at the India-based Neutrino Observatory (INO) aims to detect atmospheric muon neutrinos and antineutrinos separately in the multi-GeV range of energies and over a wide range of path lengths. While passing through the Earth, the upward-going neutrinos experience a density-dependent matter effect, which can be utilized to extract information about the internal structure of Earth. Since the Earth's matter effect modifies the neutrino oscillation patterns differently for neutrinos and antineutrinos, the capability of ICAL to distinguish $\mu^-$ and $\mu^+$ events plays an important role in observing this matter effect. Taking advantage of good angular resolution, ICAL would be able to observe about 331 $\mu^-$ and 146 $\mu^+$ events corresponding to the core-passing neutrinos and antineutrinos, respectively, in 10 years. We demonstrate for the first time that ICAL would be able to validate the presence of Earth's core by ruling out a two-layered profile consisting of only mantle and crust in fit with respect to the PREM profile in data with a median $\Delta \chi^2$ of 7.45 for normal mass ordering (NO) and 4.83 for inverted mass ordering (IO) using 500 kt$\cdot$yr exposure. If we do not use the charge identification capability of ICAL, these sensitivities deteriorate to a $\Delta\chi^2$ of 3.76 for NO and 1.59 for IO.