I am not a moose hunter, so I will just offer my opinion here FWIW.
A first hunting rifle should not be a hard kicking gun. Usually, there is some bench shooting required to sight in and that's a great place to get a flinch. Even a 308/30-06 can have too much kick to shoot sitting at the bench for some people. If he holds the gun too loosely, he can also get scoped at the bench, which my 7-08 did to me when I wasn't paying attention. BTW there was no blood, just a tap.
Many have used 7-08, 308, 30-06 and 6.5x55 on moose with good success so they should be adequate at reasonable ranges. Using premium hunting bullets is recommended, Nosler partitions and Barnes being very good bullets. A magnum calibre is certainly not required for moose hunting.
My suggestion would be to go with a Browning x-bolt, as they have a good recoil pad and get good reviews regarding felt recoil. Ask to have it bore sighted when you buy it. My calibre recommendation would be a 308, since it's very easy to find ammo and has a bit less kick than a 30-06. The 7-08 would also be a great choice. To start out, shoot lighter weight bullets or even reduced recoil loads.
First shooting should be done off-hand, not sitting, at close range targets. In this way, he can get used to the recoil and gain confidence in his ability to shoot it without fear. It won't be long before he is shooting full power loads from different positions if he has a "normal" tolerance for recoil.
Cheers