Classic discussion...
Some countries allow it (Zimbabwe, Mozambique) but several countries require .375 caliber as a minimum legal for dangerous game (Botswana, some South Africa Provinces, Tanzania, Zambia, Central African Republic); a few others .354 (Cameroon) or .366/9.3 mm (Zimbabwe for Elephant, Hippo and Buffalo); and Namibia requires 5,400 Joules for all dangerous game.
In truth:
1) The .338 with proper bullets will absolutely do the job on a lion.
2) No dangerous game will ever differentiate between a .358 (e.g. the excellent .358 Norma Magnum) a .366 (e.g. the excellent 9.3x64 Brenneke; the eminently shootable and barely sufficient - but sufficient indeed - 9.3x62 Mauser; or the newer 9.3x66 Sako), and a .375 of appropriate power (e.g. H&H, Wby, Ruger, etc. but excluding .375 Win - and the .378 Wby lives in a different world) when used with bullets of proper weight and construction adapted for the animal at hand.
3) No lion will ever differentiate between Zimbabwe's 4,300 Joules (3,170 ft./lbs.) rule that makes the .338 legal for Lion, and Namibia's 5,400 Joules (3,982 ft./lbs.) rule for all DG, that makes the .338 Win mag "technically" illegal there for a Game Department official intent on ruining your day (225 gr @ 2,800 ft./s = 3,916 ft./lbs. - for those who believe in speed for DG; and 250 gr @ 2,660 ft./s = 3,927 ft./lbs. - for those who believe in weight for DG). Not that a Lion would care either way...
My own view...
If paper requirements did not exist, my own medium-large favorite bore would be the .340 Wby that hits on DG like Thor's own hammer with 250/275 gr AFrame or 250/285 TSX; penetrates like crazy with 250 gr Peregrine monolithic solid; and shoots on PG like a laser beam with 210 gr TTSX, darn near as flat as a .257 Wby, which is no faint praise and which no .375, H&H or souped-up Wby, can do...
As it is, and as much as I love the .340 Wby, I keep it for Moose in Alaska, and pack a .375 H&H Blaser R8 barrel (although the temptation to rim it to .375 Wby is always there) to Africa because I just do not have the patience anymore to go through the arguing with the local Game Warden...