This is really a comment but I am not entitled.  There is a purely topological characterization of $n$-cubes due to J. de Groot in the Felix Hausdorff Gedenkband (1972): Topological characterization of metrizable cubes.  This specialises to one for the closed disc as requested in your question.  The condition is too intricate for me to quote it here but can be easily accessed at the review of this article (MR0348723).