Well, if the moon orbits far enough from its planet and is large enough, the answer is probably yes. However, in our Solar system that is not possible. The so-called Hill spheres¹ for the largest moons are only a few moon radii. Any object orbiting that close will be unstable, not to mention additional disturbances caused for example by other moons in the system.

It is not far-fetched to suggest that some of the distant irregular satellites orbiting giant planets may actually be binaries. The irregular satellites form families which suggests that they are fragments from larger objects.

¹) Hill sphere = the region where an object dominates its parent body. For example, our Moon is located in Earth's Hill sphere. It is only an approximation; there are no known moons that are near the border of a Hill sphere because their orbits become unstable well before that.

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Science is a way of trying not to fool yourself. The first principle is that you must not fool yourself, and you are the easiest person to fool.
-- Richard Feynman