The emergence of the Netherlands as an independent sovereign state is a process, not a single date, but the key legal milestone is 1648.
The revolt of the northern provinces (Holland, Zeeland, Utrecht, etc.) against Habsburg (Spanish) rule began in 1568. The Union of Utrecht (1579) formed a military alliance, effectively the constitutional foundation of the Dutch Republic. The Act of Abjuration (1581) formally declared independence from Philip II.
The Treaties of Münster and Osnabrück (collectively the Peace of Westphalia) ended both the Eighty Years' War and the Thirty Years' War. Spain formally recognized the independence of the Dutch Republic (Republic of the Seven United Netherlands). This is the moment the Netherlands entered the international state system as a sovereign entity.
So, while Dutch statehood has deep roots, 1648 is the definitive date of recognized sovereignty, and the state has maintained continuous legal personality since (despite regime changes).