Previous work introducing deterministic modeling of driving and dissipation to nonlinear surface gravity wave dynamics is extended to two horizontal spatial dimensions. It is shown that the wave spectrum rapidly develops into a form with two important features. First the spectral peak location is determined by the wind speed and shifts toward lower wave numbers as a function of time. This is due in part to nonlinear interactions, but it is also due to the functional form of the wind-forcing term. In addition, the spectrum rapidly develops an asymptotic power law tail in the downwind direction. The spectral exponent governing the asymptotics is sensitively dependent on the precise form of the dissipation term, and it can be tuned by adjusting that term in a quantitatively established manner. The angular dependence of the wave spectrum is also obtained. The strength and the role of the nonlinear interactions in the development of the spectral shape are studied in detail. The question of whether the wave amplitude statistics approach a Gaussian form is investigated. We find that a low-order odd moment is nonvanishing.