I like the way you first phrased the question- "why do we see it as a force?", i.e. why do we perceive acceleration. The questions about whether it really is
acceleration are more semantic.
I believe the answer is this. Take two objects, say you and the earth, which are traveling in parallel. That means that their distance isn't changing. A situation like that exists if you jump, when you're right at the top of the jump. Follow the paths, and these two parallel-moving objects collide. So either one or both accelerated, or what looked like parallel lines intersected, in which case the space is curved.
Our spacetime is only mildly curved near here. That means that it can be approximately pictured as a flat space, so long as you add some extra acceleration effects (called gravity). Our minds seem to naturally grab on to the mathematically simplest picture which does an adequate job of describing things. Flat space is much simpler than curved space, so that's how we see it.
(published on 09/02/2010)