In a word, yes. The Ottoman Empire had an Islamic legal system and specifically followed the Ḥanafi school of jurisprudence. Even the reforms of the 19th century did not entirely do away with the sharīʿa and indeed resulted in the codification of certain sections of Islamic law.

It is worth mentioning that historically Islamic states recognized a plurality of laws, which is why Jews and Christians could recourse to their own religious community's laws in certain situations. There also existed laws promulgated by Muslim sovereigns which went by various names (e.g. qanūn) and were sometimes complementary, and sometimes in tension with, the sharīʿa. The greatest Ottoman Shaykh al-Islām, Abū al-Suʿūd (Ebussuud Efendi, d. 1574), worked rather successfully to reconcile the two.