The two-dimensional Ising model with nearest-neighbour pair interactions is simulated oil the Creutz cellular automaton by using the finite-size lattices with the linear dimensions L = 80, 120, 160, and 200. The temperature variations and the finite-size scaling plots of the specific heat and the Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature. The approximate values for the critical temperature of the infinite lattice T-c = 2.287(6) T-c = 2.269(3), and T-c =2.271(1) are obtained from the intersection points of specific heat curves. Binder parameter curves, and the straight line fit of specific heat maxima, respectively. These results are in agreement with the theoretical value (T-c =2.269) within the error limits. The values obtained for the critical exponent of the specific heat, alpha = 0.04(25) and alpha = 0.03(l), are in agreement with alpha = 0 predicted by the theory. The values for the Binder parameter by using the finite-size lattices with the linear dimension L = 80, 120, 160, and 200 at T-c = 2.269(3) are calculated as g(L)(T-c) = -1.833(5), g(L)(T-c) = -1.834(3), g(L)(T-c) = -1.832(2), and g(L)(T-c) = -1.833(2), respectively. The value of the infinite lattice for the Binder parameter, g(L)(T-c) = -1.834(11), is obtained from the straight line fit of g(L)(T-c) = -1.833(5), g(L)(T-c) = -1.834(3), g(L)(T-c) = -1.832(2), and g(L)(T-c) = -1.833(2) versus L = 80, 120, 160, and 200, respectively.