Since the Moon has no atmosphere, is it possible to perform the Galileo's Leaning Tower of Pisa experiment on its surface? Has any astronaut performed anything similar?
Performing the experiment on the Moon would make it more accurate. The lack of atmosphere means there's no drag on the falling objects to interfere with the measurements.
Galileo was smart/lucky enough to pick two balls which would minimize the effect of then very poorly understood drag. There was a small discrepancy due to drag when the two landed, but Aristotle had stated things fall with a velocity proportional to their weight, so this discrepancy should have been huge. Had Galileo picked something else with more drag, the discrepancy might have been larger and the march of science might have been set back.