So what you are saying is equivalent to if I give you the set of real numbers you can find new numbers between them which is not real number. This is of cause wrong.
No. It is like saying that if you give me an arbitrary set of real numbers, I can give you back a larger set of real numbers, with additional ones between each consecutive in the original set pair. Which is of course correct.
(1) You must show that your set of great circles is analogous to the real numbers to invoke your counterargument. (2) It is not trivially clear that the real numbers come with a notion of "density" in the manner that you'd like to use it.