We have used exact numerical diagonalization to study the excitation spectrum and the dynamic spin correlations in the s = 1/2 next-next-nearest-neighbor Heisenberg antiferromagnet on the square lattice, with additional four-spin ring exchange from higher-order terms in the Hubbard expansion. We have varied the ratio between Hubbard model parameters t/U to obtain different relative strengths of the exchange parameters, while keeping electrons localized. The Hubbard model parameters have been parametrized via an effective ring exchange coupling J(r), which have been varied between 0 J and 1.5 J. We find that ring exchange induces a quantum phase transition from a (pi, pi) ordered state to a (pi/2, pi/2) ordered state. This quantum critical point is reduced by quantum fluctuations from its mean-field value of J(r)/J = 2 to a value of similar to 1.1. At the quantum critical point, the dynamical correlation function shows a pseudocontinuum at q values between the two competing ordering vectors.