The Wirtinger integral is the uniformization to the upper half plane H of the hypergeometric function defined on the complex projective line P^1. In [5] we established the transformation formulas of the Wirtinger integral for the linear fractional transformations τ → τ + 2 and τ → -1/τ with the aide of the theory of theta functions. As a corollary we obtain the transformation formulas of the Wirtinger integral for the linear fractional transformations τ → τ + 2 and τ → τ/(-2τ + 1) which are identified with generators of the principal congruence subgroup Γ(2) modulo center. These formulas correspond to the monodromy matrices of the hypergeometric function for generators of the fundamental group of P^1 minus three points. The purpose of this paper is to generalize this result, that is, we establish the transformation formula of the Wirtinger integral for a general element (a b c d) of Γ (2), which corresponds to a general monodromy matrix of the hypergeometric function.