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In this paper, we present the first formulation of the theory of nonlinear particle acceleration in collisionless shocks in the presence of neutral hydrogen in the acceleration region. The dynamical reaction of the accelerated particles, the magnetic field amplification, and the magnetic dynamical effects on the shock are also included. The main new aspect of this study, however, consists of accounting for charge exchange and the ionization of a neutral hydrogen, which profoundly change the structure of the shock, as discussed in our previous work. This important dynamical effect of neutrals is mainly associated with the so-called neutral return flux, namely the return of hot neutrals from the downstream region to upstream, where they deposit energy and momentum through charge exchange and ionization. We also present the self-consistent calculation of Balmer line emission from the shock region and discuss how to use measurements of the anomalous width of the different components of the Balmer line to infer cosmic ray acceleration efficiency in supernova remnants showing Balmer emission: the broad Balmer line, which is due to charge exchange of hydrogen atoms with hot ions downstream of the shock, is shown to become narrower as a result of the energy drainage into cosmic rays, while the narrow Balmer line, due to charge exchange in the cosmic-ray-induced precursor, is shown to become broader. In addition to these two well-known components, the neutral return flux leads to the formation of a third component with an intermediate width: this too contains information on ongoing processes at the shock. [ABSTRACT FROM AUTHOR]