Short answer: No. This is because at the surface of the moon the moon’s gravity is stronger than the earth’s gravity.

Long answer: Newton’s law of gravity says that the force of gravity between two object decreases as the distance between them increases. The moon’s surface gravity causes an acceleration towards the moon at 1.62 m/s^2. The earth’s gravity, on the surface of the moon, causes an acceleration of approximately 0.003 m/s^2 towards the earth. So, not negligible, but also not enough to cause an object to fall towards the earth. In order for the earth’s gravity and the moon’s gravity to be equal to each other, you must be about 10% of the way from the moon to the earth, or about 38,000 km from the surface of the moon. This point is called a “Lagrange point”, where gravitational forces are cancelled out to create a point of equilibrium. This Lagrange point is like balancing a pencil on your finger; one small push towards the earth causes the object to fall towards the earth, but a small push towards the moon causes the object to fall towards the moon. I hope this helps!