We consider transverse stability of line solitary waves for the KP-II equation and the 2-dimensional Benney Luke equation in a parameter regime where the surface tension is weak or negligible. Parameters $c(t,y)$ and $\gamma(t,y)$ which describe the local speed and the local phase shift of the crest of a modulating line solitary wave $\varphi_{c(t,y)}(x-\gamma(t,y))$ are described by a system of dissipative wave equations. Along with the large time behavior of these parameters and the local energy estimates for the remainder parts of solutions around line solitary waves, we obtain a bound for the $L^2$-norm or the energy norm of the remainder part and prove nonlinear stability of line solitary waves.