Solar eclipse is the eclipse of the Sun by the Moon.
You need a Lunar eclipse to see the shadow of the Earth on the Moon.
A lunar eclipse occurs when the Moon passes directly behind the Earth into its umbra (shadow). This can occur only when…en.wikipedia.org
The Earth shape projected on the Moon is not so obvious in one observation, but the measurement of the shadow can be used to calculate the size of the Earth, compared to the size of the Moon, provided the Sun is known to be much farther.
I guess to prove the roundness of the Earth, the easiest experiment a famous person can do is to make his/her friends from all over the world send the text messages every hour during, let’s say, 24 hours and write where they observe the Sun — that way it can be followed “circling” the Earth. The only thing is, they should not lie.
Regarding proving the “shape of sphere” in the lab, the problem is that the gravitation is the weakest force of all. AFAIK, only analogies are possible, not any direct proof.