I don't see how. A "force between two bodies with a magnitude proportional to the product of their masses and inversely proportional to the square of the displacement between them" seems no different in principle from a "force between two bodies with a magnitude proportional to the product of their electric charges and the inversely proportional to the square of the displacement between them".

My point is that it really seems like "gravity" is not a force, it's an acceleration, but there is something you could call "force due to gravity" that you reverse-engineer from the known acceleration, and that means you need to multiply by the mass. Clearly different masses will just cancel, so the resulting acceleration is the same.

We're talking Newtonian physics, a model of the world where an acceleration is the result of an unbalanced force applied to a mass. You don't have to "reverse-engineer it from the known acceleration", you can calculate it as Gm₁m₂/r².

I'm fine with saying "force due to gravity" or even "gravitational force". Which is what you're describing in your first sentence, and I have no disagreement with that. "Force of gravity" starts to sound a little off, and I bet if I ask "what is gravity at Earth's surface?" I'll get back "9.8 m/s^2", which is an acceleration not a force.

You can equally describe the magnitude of the gravitational field as 9.8 Nkg⁻¹.