A few comments: I'd say it's an overstatement to say that physics does or can us "what" gravity is. But good physics will tell us how gravity acts. If two different models of gravity give the same predictions, physics doesn't have any definite way of choosing which model to use. It typically doesn't make any direct difference, but it may make a difference when it comes time to expand the theory to cover more and broader situations. It's generally impossible to tell in advance which form of a theory will be the most amenable to expansion, however.
I believe "Exploring Black Holes" would be the best source for explaining how gravity comes out of the principle of maximal aging. I don't have a copy of the book, unfortunately, to give any quotes, but there are a few tidbits on Taylor's website,
http://www.eftaylor.com/leastaction.html. This is the closest I can think of to what I would guess the OP may be asking, though I suspect it's not a really great match. It's close enough to be interesting to him, perhaps, but I'm not sure if there is a good elementary treatment of it out there.
There are some other interesting models of gravity that don't initially involve geometry at all, such as Straumann's "Reflections on gravity". Interestingly, geometry comes out in the end. I suspect the paper is too advanced for the OP, however. I'll give the link anyway -
http://arxiv.org/abs/astro-ph/0006423.
My personal view is that trying to reduce gravity to "just a force" winds winds up a lot like the fabled Procrustes, who would make people fit an iron bed by stretching their bodies in a rack, or cutting their legs down to size, rather than adjusting the size of the bed.