No, your Warlock would not get the chance to make a Ranged Spell Attack roll on following turns due to the specific wording of the spell. He would, however, maintain Concentration on the spell.
He cannot try again due to it not being stated within the spell's effects, as it is stated in a similar spell, Vampiric Touch:
Until the spell ends, you can make the attack again on each of your turns as an action.
Due to the rules of Duration: Concentration (PHB, pg. 203), the Warlock can maintain Concentration on the spell after the initial attack roll miss.
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends.
The following factors break concentration: Casting another spell that requires concentration, Taking damage, Being incapacitated or killed
As none of those requisites are met by the ranged spell attack missing, the Warlock may maintain concentration on his Witch Bolt spell but will not be able to apply any of the spells effects (explicitly stated as "On a hit") or make any additional ranged spell attack rolls (as it is not stated within the spells effects separate to the "On a hit" effects).
In conclusion, your Warlock may choose to maintain his concentration on the spell, but gains none of its effects (or any additional ranged spell attacks) for the duration he maintains concentration on it.