The first thing to establish is whether or not the moon would fall or not. Mach's principle suggests it would fall, because his principle says that rotation is only possible when there are distant bodies to rotate relative to. When Einstein was formulating General relativity, he considered Mach's principle, but ultimately Einstein quietly rejected the principle and did not incorporate the idea into General Relativity. It can be seen that the Schwarzschild solution for a non rotating black hole and the Kerr solution for a rotating black hole do not require distant background bodies to establish whether rotation is occurring or not. They are both vacuum solutions. Newton's considered the idea of rotating a bucket of water to argue for the notion of rotation being absolute. When the water in the bucket is rotating it forms a curved top surface and climbs up the walls due. Newton attributes the reaction of the water to the absolute nature of rotation and centrifugal force. Indeed it is hard to imagine that if instead of rotating the water in the bucket, we rotated everything in the Universe beside the water in the bucket, that the only result would be the curvature of the water in the bucket.

The question states there is nothing in the universe besides the planet and the moon but it does also state there are observers. Assuming these are human like observers, they would eventually notice that they fall back down when they jump into the air and deduce there is some sort of force attracting them downwards. Eventually they would wonder why the moon does not fall straight down. If they tie a rock to a rope and swing it about heir heads they would eventually figure out the existence of centrifugal forces. In the absence of rocks and rope, they could hold the hands of a small observer and swing the child around around themselves in circles and notice the child rises up off the ground. If there is a smart one amongst them like our Newton, they would eventually figure out that centrifugal force can oppose gravitational force and this could explain why the moon does not fall straight down, if they assume the planet and the moon must be rotating at the same rate as each other.

The first thing to establish is whether or not the moon would fall or not. Mach's principle suggests it would fall because his principle says that rotation is only possible when there are distant bodies to rotate relative to. When Einstein was formulating General relativity, he considered Mach's principle, but ultimately, Einstein quietly rejected the principle and did not incorporate the idea into General Relativity. It can be seen that the Schwarzschild solution for a non-rotating black hole and the Kerr solution for a rotating black hole do not require distant background bodies to establish whether rotation is occurring or not. They are both vacuum solutions. Newton considered the idea of rotating a bucket of water to argue for the notion of rotation being absolute. When the water in the bucket is rotating, it forms a curved top surface and climbs up the walls. Newton attributes the reaction of the water to the absolute nature of rotation and centrifugal force. Indeed, it is hard to imagine that if instead of rotating the water in the bucket, we rotated everything in the Universe besides the water in the bucket, the only result would be the curvature of the water in the bucket.

The question states there is nothing in the Universe besides the planet and the moon, but it does also state there are observers. Assuming these are human-like observers, they would eventually notice that they fall back down when they jump into the air and deduce there is some sort of force attracting them downwards. Eventually, they would wonder why the moon does not fall straight down. If they tie a rock to a rope and swing it about their heads, they would eventually figure out the existence of centrifugal forces. In the absence of rocks and rope, they could hold the hands of a small observer, swing the child around themselves in circles, and notice the child rise up off the ground. If there is a smart one amongst them like our Newton, they would eventually figure out that centrifugal force can oppose gravitational force, and this could explain why the moon does not fall straight down if they assume the planet and the moon must be rotating at the same rate as each other.