BUDAPEST—Hungary's central bank is at the center of an increasingly acrimonious debate about the direction of economic policy in Central Europe's most indebted country, reflecting its persistent vulnerability on financial markets and risking a sharp weakening of its currency.

The National Bank of Hungary on Monday issued a sharply worded statement saying new bank-tax measures Prime Minister Viktor Orban's populist government announced last Friday could threaten the stability of the national economy.

The...