The Stokes wave problem in a constant vorticity flow is formulated via conformal mapping as a modified Babenko equation. The associated linearized operator is self-adjoint, whereby efficiently solved by the Newton-conjugate gradient method. For strong positive vorticity, a fold develops in the wave speed versus amplitude plane, and a gap as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself, enclosing a bubble of air. More folds and gaps follow as the vorticity strength increases further. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely, while a fluid disk in rigid body rotation at the ends of the gaps. Touching waves at the boundaries of higher gaps contain more fluid disks.