Radiation: absolutely yes.
But Gamma radiation is a bad fit.
The atmosphere quite effectively blocks Gamma and X-ray radiation. Even ultrahigh energy photons are very well blocked by the atmosphere. By the time the radiation at groundlevel becomes dangerous, you are in much more danger due to the atmosphere above you becoming red hot+ from absorbed energy.
But: the atmosphere is only middling effective at stopping true Cosmic Rays. i.e. relativistic speed particles, mostly protons.
Actually the upper atmosphere does stop the incoming particles quite well. Unfortunately, the act of stopping them is like stopping a bullet by having it hit a porcelain cutlery set. You stop the bullet, but you are left with a huge number of shards, splinters and debris that now carry a part of the energy with them.
If your disaster event around the Moon released not Gamma but high energy particles, then yes the Earth could suffer.
Of Course, it would have to be an event of truly mindboggling magnitude, as the energy release will be non-directional, and the Earth covers only 1/32000th of the visible sky from the Moon's distance. Then, any energy that reached Earth is spread over the entire facing side of the Earth, some 250 million square kilometers.
As for "Instantly Kill"
If you really need to kill everything instantly then heat pulse is your only option. This is not a good idea, as I believe you want the other side of the Earth to survive? Instantly immolating half of Earth will kill the whole Earth within hours, as the heat will cause a shockwave to circle the Earth, scouring the surface all the way.
If you just need to kill everyone on the surface soon, as in within an hour or so, that is much more achievable.
Acute radiation poisoning, if severe enough, will cause cell membranes to depolarize, causing the nervous and respiratory systems to collapse. The dosage require is truly monstrous though, on the order of 50 time normal "lethal" dose. Figure 200 Sievert or more.
If you can wait a month or so for the dieoff, you only need about 8-10 Sievert. This is the normal "lethal, no chance" radiation dose that kills in several days to about a month.
No matter what the dosage, as the radiation is only in the form of high energy particles, the residual radiation will very rapidly fade down to reasonable levels. You would not permanently contaminate the whole planet unless your radiation pulse also included a lot of Neutrons. (In which case, bye-bye whole planet. for many millions of years)
Except for residual radioactivity, the Ocean will be unaffected. Water is an amazing radiation shield to all kinds of radiation. The energy of radiation needed to sterilize the ocean depths would achieve that sterilization by explosive boiloff of the water, before the radiation itself is an issue.