Final answer:
Greece achieved independence from Turkish rule in the 1820s with the aid of Britain, France, and Russia, which sunk an Ottoman fleet in 1827. Russia declared war on the Ottomans in 1829, and Greece was officially recognized as independent in 1833, an act contrary to the Holy Alliance's objectives.
Step-by-step explanation:
The nation which achieved independence from Turkish rule with assistance from Britain, France, and Russia, acting against the Holy Alliance's principles, was Greece. After a series of uprisings that began in 1821, and the European public's sympathy, triggered largely due to Greece's historic significance as the cradle of European culture, the Great Powers decided to intervene. The combined fleet of Britain, France, and Russia decisively defeated an Ottoman fleet in 1827, and with further military support, notably Russia's declaration of war against the Ottomans in 1829, Greece secured independence, officially recognized in 1833.
The Romantic movement's artists and writers, inspired by Greece's struggle, contributed to the European public's backing for the Greek cause. Although this was in contradiction to the Holy Alliance's aim of preventing such nationalistic uprisings, the Great Powers' own geopolitical interests led them to support Greek autonomy, striving to enhance their influence in the Eastern Mediterranean and to manage the decline of the Ottoman Empire strategically.
The decline of the Ottoman Empire marked a significant transition in geopolitics, leading to the emergence of new nations from its former territories and setting the stage for both Greece's independence and the later Balkan states' push for autonomy.