The critical behavior of a two-dimensional (2D) system of hard disks, each disk bearing an Ising spin and interacting via Ising-like interactions, is studied near the second-order phase transition from the paramagnetic-to-ferromagnetic fluid phase by Monte Carlo simulations combined with a finite-size scaling analysis and a single histogram technique. The critical line is located and values of critical exponents along this line are determined. The continuous variation of the Binder's reduced fourth-order cumulant at the critical point and of the critical exponent nu with disk density is observed and a linear relation between these quantities is found. Ratios of critical exponents gamma/nu and beta/nu are found to be the same as those in the 2D Ising lattice model at all fluid densities. A statistical analysis of Voronoi diagrams generated for typical particle configurations along the critical line is performed. The variation of parameters characterizing Voronoi diagrams at the critical point with density is compared to the variation of critical exponents with density.