We know from the idea of wave-particle duality that electromagnetic waves are photons which have the property of momentum. It stands to reason that gravitational waves would have similar duality. Would the particle form of gravitational waves carry momentum the same as a photon does? If so, would this give gravitational waves a kind of repelling effect pushing away atoms they hit much like a solar-sail?
Yes. Gravitational waves carry energy, momentum, and angular momentum, just like electromagnetic waves. This is classical field behavior; you don’t have to quantize either field to have this effect. If you do, you can understand the energy, momentum, and angular momentum in terms of gravitons and photons, at least in the weak-field limit.
The gravitational radiation pressure on objects such as solar-sails is completely negligible.
The main effect of momentum being radiated gravitationally is black-hole merger recoil, which can eject the merged hole from its galaxy!