Ira Gessel used transitive tournaments in graphs to prove Vandermonde’s determinant identity: http://onlinelibrary.wiley.com/doi/10.1002/jgt.3190030315/abstract This proof certainly has some geometric flavour although not in the initial sense of the question.
P.S. The historical process that led to the worldwide adoption of the denomination "Vandermonde determinant" is studied in http://arxiv.org/abs/1204.4716 (A case of mathematical eponymy: the Vandermonde determinant, by Bernard Ycart).
Interestingly, $3\times 3$ Vandermonde determinant turns out to be (up to the square root) the correct physical variable of the odderon due to modular invariance of the odderon: http://arxiv.org/abs/hep-th/9604162 (The Odderon and Invariants of Elliptic Curves, by Romuald A. Janik).