The numerical simulation of incompressible fluid-structure interaction systems with loosely coupled schemes is a delicate problem. Indeed, the splitting method must both be stable for the full nonlinear system and have sufficient accuracy to be of use in practice. In the case of the coupling of an incompressible fluid with thick-walled solids the error analyses reported in the literature are limited to 1/2-order accuracy in time. The objective of the present work is to show two important extensions of the analysis of the Robin-Robin loosely coupled scheme recently reported in [Numer. Math., 151(4):807–840, 2022]. First, we give a formulation of the scheme in a general non-linear setting and prove its unconditional energy stability. Then we show that nearly-optimal accuracy can be achieved in the linear case. These theoretical findings are illustrated in a series of numerical examples.