Advanced imaging techniques generate large datasets that are capable of describing the structure and kinematics of tissue spreading in embryonic development, wound healing, and the progression of many diseases. Information in these datasets can be integrated with mathematical models to infer important biomechanical properties of the system. Standard computational tools for estimating relevant parameters rely on methods such as gradient descent and typically identify a single set of optimal parameters for a single experiment. These methods offer little information on the robustness of the fit and are ill-suited for statistical tests of different experimental groups. To overcome this limitation and use large datasets in a rigorous experimental design, we sought an automated methodology that could integrate kinematic data with a mathematical model. Estimated model parameters are represented probability density distributions, which can be constructed by implementing the approximate Bayesian computation rejection algorithm. Here, we demonstrate this method with a 2D Eulerian continuum mechanical model of spreading embryonic tissue. The model is tightly integrated with quantitative image analysis of different sized embryonic tissue explants spreading on extracellular matrix (ECM). Tissue spreading is regulated by a small set of parameters including forces on the free edge, tissue stiffness, strength of cell-ECM adhesions, and active cell shape changes. From thousands of simulations of each experiment, we find statistically significant trends in key parameters that vary with initial size of the explant, e.g., cell-ECM adhesion forces are weaker and free edge forces are stronger for larger explants. Furthermore, we demonstrate that estimated parameters for one explant can be used to predict the behavior of other explants of similar size. The predictive methods described here can be used to guide further experiments to better understand how collective cell migration is regulated during development and dysregulated during the metastasis of cancer.