It depends on exactly what you mean.
There have been British/ English naval forces since the 10th century, but they were organised and administered as the newly-named Royal Navy by Henry VIII in the 16th century.
Royal Navy was created in 1707.
Royal Netherlands Navy was created in 1488.
Royal Saudi Navy was created in 1960.
Royal Lao Navy was created in 1955.
Ships of the Royal Navy was created in 1969.
Royal Cambodian Navy was created in 1953.
Royal Norwegian Navy was created in 1814.
Royal Brunei Navy was created in 1965.
Royal Australian Navy was created in 1901.
Royal New Zealand Navy was created in 1941.
Royal Dano-Norwegian Navy was created in 1509.
Royal Navy Submarine Museum was created in 1963.