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We present a numerical study of essentially nonlinear dynamics of surface gravity waves on deep water with constant vorticity using governing equations in conformal coordinates. The dispersion relation of surface gravity waves on shear flow is known to have two branches, one of which is weakly dispersive for long waves. Weakly nonlinear evolution of the waves of this branch can be described by the Benjamin-Ono equation, which is integrable and has soliton and multi-soliton solutions. Currently, the extent to which the properties of such solitary waves obtained within the weakly nonlinear model are preserved in the exact Euler equations is unknown. We investigate the behaviour of this class of solitary waves without the restrictive assumption of weak nonlinearity by using the exact Euler equations. The evolution of localized initial perturbations leading to the formation of single or multiple solitary waves is modeled, and the properties of finite-amplitude solitary waves are discussed. We show that within the framework of the exact equations, two-soliton collisions are almost elastic, but in contrast to solutions of the Benjamin-Ono equation the waves receive a phase shift as a result of the interaction.