That depends a lot of your definition of what you call "maritime imperialism". The Ottoman Empire was a first rate naval power in the Mediterranean Sea well into the 17th century. In the 16th century they were actually strong enough to support operations deep in the Western Mediterranean like sending an expeditionary force to Andalusia to support the Morisco uprising in 1570. The Ottomans' enormous galley fleet was frighting enough to bring the Christian powers of the Mediterranean (Spain, Venice, Genoa among others) together in the Holy League that managed to beat them at Lepanto in 1571. While their exploits in the Indian Ocean are much less known commonly, they did have a sizable presence there too, at least until the end of the 16th century.

I guess when you are using the term "maritime imperialism", you mean to ask why the Ottoman Empire did not develop a concept of empire overseas, an imperial strategy either focussed on exploiting superior naval technology like ocean-going ships to conquer key points (harbors and coastal fortresses) to control trade networks (i.e. like Portugal) or on acquiring large territories in technologically less developed regions to extract ressources and/or settle them (like Spain or Great Britain later on).

But to understand geopolitical strategies like these we need to understand first that the colonial undertakings of Western European states were not always concerted or coherent efforts and secondly that they were all born out of a very different set of needs and interests which they pursued. Without direct access to the highly valued goods of the South Asian (spices) and East Asian markets (silk, china) Christian Europe had always been dependent on intermediary forces either by land (through Central Asia like the silk road) or by sea (through the Indian Ocean trade network which was connected to the Mediterranean mainly through Egypt), with the latter being the most important by far in the Early Modern Era. The geographic position of Western Europe created the unique incentive to go out actively looking for new trade routes to cut out the middle man. (Portugal succeeded first in this by reaching India in 1498 by going around Africa, Spain tried a different approach and discovered the Americas by accident which in turn changed history in a way not foreseen by anybody at the time.) The Ottoman Empire on the other hand had no incentive to do so. It was the middle man. Well, it was the middle man at the latest since the Ottoman conquest of the Mamluk Sultanate of Egypt in 1517 which led the Empire to control of the single most important Eurasian trade hub.

The Ottoman Empire altogether was in a very different position than its Western European rivals as it actually controlled the sought-after trade routes, thus its maritime strategy was naturally different from its opponents. With that being said, what did the Ottomans do to counter the new European (that means Portuguese) presence in the Indian Ocean since the early 16th century? Well, they tried a lot of things with varying success. Ottoman fleets launched from the Red Sea (the Ottoman arsenal was located in Suez) did pose a significant threat to Portuguese shipping and between 1517 and 1589 there was a lot of back and forth fighting with the Portuguese raiding deep into the Red Sea at times, and the Ottomans raiding the Portuguese-controlled parts of India and East Africa at times and even sending an expedition to Sumatra. For close to a century this was by all means a fight with open end.

In the mind of Ottoman policy makers however (think changing viziers, governors, sultans etc.) the Indian Ocean was always one of many different theaters to focus on. They had a long border on the Balkans to fight the powerful Habsburgs and Poland-Lithuania, they had two other seas to manage (the Med. and the Black Sea), and they shared a long border in the east with another strong opponent as well -- Safavid Persia, with whom the Ottomans fought several wars. After the debacle of Mombasa in 1589 (where an Ottoman expeditionary force was lost entirely) which was not the first of a series of unfruitful enterprises, Ottoman presence in the Indian Ocean was diminished partly by own choice. Being still in control of Egypt and the Red Sea, the Empire could afford to let go and let trade flow on its own and collects its share. (But by now the world had thoroughly changed anyway.)

It remains to argue whether the Ottoman had serious imperial ambitions for the Indian Ocean world. Giancarlo Casale in The Ottoman Age of Exploration argues that by presenting themselves (in letters, declarations, embassies etc.) to the South Asian Muslim world as the custodians of the Holy Cities and Caliphs after 1517, the Ottomans did conceive the idea of an universalist empire that holds sway over far-off client states overseas not unlike the Portuguese and the Spanish did. He makes a good case, I think. In the 16th century several Muslim states, notably the Sultanate of Aceh (part of modern Indonesia) and the Sultanate of Gujarat (in India), did accept nominal overlordship of the Ottoman caliph and apparently the khutbah, the Islamic sermon, was read out in the name of Ottoman rulers from the Middle East to South Asia, indicating a certain amount of influence. The extent of the influence remains debatable.