We explore the physics of the gyro-resonant cosmic ray streaming instability (CRSI) including the effects of ion-neutral (IN) damping. This is the main damping mechanism in (partially-ionized) atomic and molecular gas, which are the primary components of the interstellar medium (ISM) by mass. Limitation of CRSI by IN damping is important in setting the amplitude of Alfven waves that scatter cosmic rays and control galactic-scale transport. Our study employs the MHD-PIC hybrid fluid-kinetic numerical technique to follow linear growth as well as post-linear and saturation phases. During the linear phase of the instability -- where simulations and analytical theory are in good agreement -- IN damping prevents wave growth at small and large wavelengths, with the unstable bandwidth lower for higher ion-neutral collision rate $ u_{rm in}$. Purely MHD effects during the post-linear phase extend the wave spectrum towards larger $k$. In the saturated state, the cosmic ray distribution evolves toward greater isotropy (lower streaming velocity) by scattering off of Alven waves excited by the instability. In the absence of low-$k$ waves, CRs with sufficiently high momentum are not isotropized. The maximum wave amplitude and rate of isotropization of the distribution function decreases at higher $ u_{rm in}$. When the IN damping rate approaches the maximum growth rate of CSRI, wave growth and isotropization is suppressed. Implications of our results for CR transport in partially ionized ISM phases are discussed.
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