First, we prove that a Ritt operator T on a Hilbert space admits a bounded \(H^\infty \) -functional calculus if and only if it is similar to a contraction, which is also equivalent to being similar to an operator whose numerical range is contained in the closure of a Stolz domain \(B_\gamma \) . Next, on UMD spaces, we characterize Ritt operators with a bounded \(H^\infty \) -functional calculus through an isometric dilation property. This characterization is then simplified in a striking manner when the underlying space is either an \(L^p\) -space or a quotient of a subspace of an \(L^p\) -space.