Throughout his voyages to the americas columbus steadfastly believes he has discovered islands off the coast of what country?

The correct answer is - India.

Columbus went on a mission to discover a sea route to India from the opposite side, or rather to move towards west so that he comes out on the eastern part of India. So when he discovered the islands in the Caribbean he had no idea that those islands are a part of a totally new world unknown to the Europeans (apart from the Vikings), and he thought that those islands must be islands situated east of India, so that is why he also gave the name Indians to the native populations he encountered.