We report localized and unidirectional nonlinear traveling edge waves in a 2D mechanical (phononic) topological insulator consisting of a collection of pendula with weak Duffing nonlinearity connected by linear springs. This is achieved by showing theoretically that the classical 1D nonlinear Schrödinger equation governs the envelope of 2D edge modes. The theoretical predictions from the 1D envelope equation are confirmed by numerical simulations of the original 2D system for various types of traveling waves and rogue waves. As a result of topological protection, these edge solitons persist over long time intervals and through irregular boundaries. The existence of topologically protected edge solitons may have significant implications on the design of acoustic devices.