We introduce a geodesic synthetic control method for causal inference with panel outcomes that are random objects in a geodesic metric space. Examples include distributions, compositions, networks, symmetric positive-definite matrices, trees and functional data, among other data types that require a geometry beyond Euclidean vector spaces. The proposed method replaces Euclidean weighted averages by weighted Fréchet means and defines treatment effects as geodesics connecting untreated and treated potential outcomes. We develop a causal model with stochastic perturbations for object-valued untreated outcomes, establish consistency of the estimated weights for the perturbed population target, derive a bound on post-treatment prediction error, and give sufficient conditions for consistent recovery of the untreated counterfactual. For uncertainty quantification, we propose an intrinsic metric conformal calibration procedure that yields prediction regions for untreated counterfactual objects, confidence sets for geodesic treatment effects, confidence intervals for their magnitudes, and global no-effect tests. The method is illustrated through simulation studies for networks and symmetric positive-definite matrices, and through applications to employment composition changes following the 2011 Great East Japan Earthquake and the impact of abortion liberalization on fertility patterns in East Germany.

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