BERLIN—The European Union is facing the worst crisis since its inception as the coronavirus deepens long-running tensions between the bloc’s south, hardest hit by both the pandemic and its economic fallout, and the financially stronger north.

Whether and how this clash, crystallized in the quarrel between Germany and Italy, gets resolved in the coming weeks is likely to determine the long-term viability of the entire European project—including the future of the common currency. If unresolved, that fight will continue to poison...