Curve on a sphere determined by geodesic curvature

Question

Is it possible, given an arc-length parametrized curve $\gamma : [0, 1]\to S^2$, and given $\gamma(0), \gamma'(0)$, to show that $\gamma$ depends only on its signed geodesic curvature, and if so is there a "nice" explicit expression for this? Or does one need to work with "higher" derivatives of the curve to determine its properties? It seems like an existence theorem for solutions to an ODE may suffice here, though I'm not sure how to go about implementing this.