For the second time, Iceland's president vetoed a bid by the island nation's Parliament to repay the U.K. and the Netherlands more than $5 billion lost by depositors in Iceland's epic 2008 banking collapse—sending the matter to a referendum by a deeply skeptical public and complicating the country's application to join the European Union.

The dispute over Icesave—the online arm of a failed Iceland bank that took deposits from British and Dutch savers—has percolated for more than two years, reflecting the Icelandic people's...