I'd think dwarves, of anyone, would be able to sustain and even thrive in a D&D culture without magical healing. They live underground and despite all the hideous monsters that also live underground, a dwarven complex has got to be the best defended place on the planet. So they don't have to worry a lot about wandering monsters coming in to savage farmers and things like that.
They'd develop some advanced healing techiques, perhaps even some form of healing potion made from minerals. Dwarves are great with machines and the body is just a more complex machine; they'd figure out anatomy before almost anyone else save druids.
Or, they could go the ancestor-worship route to get access to 'divine' spells. Dwarves stereotypically revere their lineage; this just takes it further.
Or, they create a form of arcane magic that can heal, via tapping the positive energy plane. It might even be a very narrowly focused PrC that only gets healing spells.