Your hand is no longer in contact with the rock, therefore there is not way for your hand to exert a force on the rock during its "flight". In that case, the only type of force left that could be exerted on the rock is gravitational force. Since you are on the moon, there is no air around you for friction (from the air) to act on the rock while in motion.
A common source of confusion for students about this topic is because tossing a rock results in a curved path, and it is strange to think that the force of gravity is the only thing acting on the rock in a curved path just like how the force of gravity is the only thing acting on a rock if you were to just let is fall straight down. In the curved path example, your hand is giving the rock initial velocities in the horizontal and upward directions. The horizontal velocity, since there is no force acting in the horizontal direction (gravitational force is an up/down force), will continue as-is for the entire duration of the flight, giving the path its parabolic shape. Compare this to if you were to toss a rock straight up - it would fall straight down without horizontal movement because there is no horizontal force to give it velocity in the side to side directions.
The rock is affected by the gravity of the moon that pulls it down. It is affected by its own inertia which pushes it forward and resists change in its motion and direction. The astronauts throw also pushes it forward. It is also slowed down by the little friction on the moon.
When a rock is tossed on the moon it follows a curved path. The only force acting on the rock continuously is the gravitational force due to the gravitational acceleration of the moon acting on its mass. There are no other force on it. The toss given to the rock the provides only an initial velocity to it which progressively deterioarates (due to the gravitational force) till it reaches a certain height and then it returns to the ground. But throught its motion it has a horizontal component of the velocity unaffected by gravity which remains same till it falls to the ground. But the vertical velocity reduceses continously till it becomes 0 and then the direction of velocity changes and increases towards the ground . Thus the gravitational force is the only force acting on the rock constantly pulling it towards the moon toll the it lands on moon.The path is determined by the initial velocity, the direction of the toss and the continuous gravitational acceleration.
When the rock is tossed by the astronaut on the Moon, he applies some force to the rock which sets the rock in motion. Once the rock leaves the hand of the astronaut, the force applied by the hand of the astronaut ceases. But the rock has acquired some motion. However the gravitational force of the Moon will start acting on it, and because of this its path of motion and speed will change as it travels which otherwise would have been uniform motion in a straight line. The stone will continue to move in this fashion till it hits the surface of the Moon under the influence of Moon's gravity.
Unlike the Earth, here is no air or gas around the moon. Therefore, there will be no force due to air resistance acting on the moving stone.