PC casts hex (PHB, p. 251) on an enemy. Then he casts witch bolt (PHB, p. 259) and succeeds. Next turn, he automatically deals 1d12 lightning damage from witch bolt. Does hex's 1d6 necrotic damage also apply? He didn't roll, but it's still an attack.
No, Witch Bolt's ongoing damage is not an "attack"
As per the basic rules, there's a simple rule for determining whether or not something is an "attack":
If there's ever any question whether something you're doing counts as an attack, the rule is simple: if you're making an attack roll, you're making an attack.
And Hex is only invoked on attacks that hit:
Until the spell ends, you deal an extra 1d6 necrotic damage to the target whenever you hit it with an attack.
Witch Bolt causes an attack to take place when the spell is first cast - since affecting the creature requires a ranged spell attack roll - but the ongoing damage you can do on subsequent turns as a result is not an attack. It just happens, with no roll required, and the spell's description doesn't call it an attack. Hex would only grant bonus damage once to the initial attack that Witch Bolt makes...
... if it weren't for the fact that Hex and Witch Bolt are both concentration spells, so you can't actually have both active at the same time anyway - as soon as you cast Witch Bolt, Hex would end, unless you somehow have a way to concentrate on two spells at once!
Note that the thematically similar Hexblade's Curse ability, available to Hexblade warlocks, has a different wording:
You gain a bonus to damage rolls against the cursed target.
This ability doesn't require an attack (nor is it a spell which requires concentration) - it simply adds to any damage roll you make against the target. A Hexblade who cast Witch Bolt at such a cursed target would get to deal bonus damage due to the curse both on the initial hit and each subsequent damage roll they made.