Your fundamental error rises from defining an arc as "a portion of the circumference of a circle". Instead, an arc is simply a portion of a curve. Dictionary.com, for instance, defines an arc (in the geometrical sense) as "any unbroken part of the circumference of a circle or other curved line." And this site refers to arcs as being parabolic or hyperbolic, and here is a discussion of arcshere is a discussion of arcs that talks about creating an arc with a Bezier curve.
What all these uses have in common is the description of a curved path. King's imagery, as many good examples do, calls to mind at least two entirely different phenomena, and applies both to the same point.
First, a geometric curve. King is describing the development of moral justice, and this is hardly a short or straight path. How to characterize such a path? Since "straight line" is not on the table, "arc" works splendidly. Consider an alternative - "The wiggly, squiggly path of the moral universe" simply doesn't have the resonance that "arc" confers.
Second, as Joel Grant points out, the phrase "story arc" seems an entirely apropos connection, as the development of moral justice is readily seen as a long, convoluted story.