The Ottomans were said big caliphate, and they didn’t really go for the whole settler colony thing. They were more interested in finding trading partners (if things had gone differently and the Native American empires had not been conquered, it’s possible Ottoman trade envoys might have shown up there eventually).

However, the European age of exploration was more than just about the Americas; from the Ottoman point of view a much more direct threat was the Portuguese circumnavigation of Africa that opened up India and the spice islands to direct trade. Up until that point the Egyptian navy dominated the Red Sea and Egyptian-based traders had a huge presence in Indian ports, and they were making a lot of money off of supplying goods to Europe.

The trade passed to the Ottomans in 1517 when they conquered Egypt, and for them trying to protect that trade from Portugal was a much more lucrative cause than exploring a continent half a world away, whose value was not yet proven, and required them to contend with not one but two enemies (Portugal, and Spain, whose navy controlled access to the Strait of Gibraltar, through which Ottoman ships would have to pass on the way to the Americas).

Giancarlo Casale’s book The Ottoman Age of Exploration details the thinking at the Ottoman court in the 16th century. It’s worth a read.