You might want to consider No Endor Holocaust. What happens when the Death star explodes near Yavin and the moon Yavin IV, and what happens when the Second Death Star explodes near Endor.
Suppose the bomb explodes and vaporizes a sphere of rock 1 kilometer in diameter. The average distance between the Moon and Earth is 384,399 kilometers. So when some of the vaporized rock reaches Earth the sphere containing the rock should have a volume about 5.6799 X 10 to the sixteenth power times the volume it originally had and thus the density will be very much less than it had before being vaporized. But its speed will be very great.
But if all of the vaporised rock has a narrow range of speeds, it won't expand as a sphere but as a comparatively narrow shell. Thus it will have a much smaller volume than the sphere calculated above and a higher density.
Someone might want to calculate the density of such a shell if it is 1,000 kilometers thick and if it is 1 kilometer thick, for example.
How dense does the expanding shell of vaporized rock have to be to do damage when travelling at a high speed? I don't know.
Triton, the large moon of Neptune, has a very, very, very thin atmosphere. It has a surface pressure of 14 microbars - one 70,000th as dense as Earth's atmosphere. But that is dense enough for clouds to form and for winds in the atmosphere to have detectable effects.
Thus It is possible that the vaporized rock from the atomic blast could have no detectable effects on Earth or wipe out all life on Earth. And it would take calculations to show how much effect there would be.
Back in the late 1950s the USA was considering a plan to send an atomic bomb to the Moon and explode it on the surface, Project A119. The plan would have used a W25 warhead with a 1.7 kiloton yield. Apparently none of the scientists calculated any danger to Earth because that wasn't the reason why the project was dropped.
Of course the plan was to detonate the bomb near the terminator and raise a dust cloud that would be illuminated by the Sun and visible from Earth, while you plan to detonate your bomb near Mare Tranquillitatis, closer to the center of the Moon and making Earth the target for more of the debris. And you plan to fictionally detonate a bomb about 58,823.529 times as powerful.