The second-order phase transition in the two-dimensional (2D) classical Coulomb gas of half-integer charges on a square lattice is investigated by using Monte Carlo simulations. Based on the finite-size scaling analysis,we estimate the second-order phase transition temperature Tc and the static critical exponents β and v with a new numerical analysis method. More precise critical temperature Tc = 0.1311(2) and critical exponents β/ν = 0.1152(12) and ν = 0.857(15) are obtained. The estimated value of ν indicates that the charge lattice melting transition is different from the pure 2D Ising transition.