If one's inner monologue stops forever, is one necessarily an arahant? Question in the title. I'm looking for answers citing either canonical scripture or the position of any particular school of Buddhism.

The inner monologue in the question could also be read as mental chatter; mental noise; superfluous thoughts; extraneous thoughts; or the  unnecessary thoughts that normal people have every moment.

And I mean when the inner monologue has stopped for good; that is, the case when it never comes back.

Another way to read this question might be, Do arahants have an inner monologue?
Thoughts are one of the six sense objects; there is no reason to think that they stop when one becomes an arahant. It is quite clear that both the Buddha and arahants did indeed have thoughts after becoming enlightened. E.g.:


  And the knowledge and vision arose in me: ‘Āḷāra Kālāma died seven days ago.’ I thought: ‘Āḷāra Kālāmaʹs loss is a great one. If he had heard this Dhamma, he would have understood it quickly.’
  
  -- MN 26 (Bodhi, trans)


What does stop forever is uddhacca or restlessness.


  Restlessness (or agitation) has the characteristic of disquietude, like water whipped up by the wind. Its function is to make the mind unsteady, as wind makes a banner ripple. It is manifested as turmoil. Its proximate cause is unwise attention to mental disquiet.
  
  -- Bhikkhu Bodhi, A Comprehensive Manual of Abhidhamma
  
  In the case of the defilements, [false] view and uncertainty are eliminated by
  the first knowledge. Hate is eliminated by the third knowledge. Greed, delusion,
  conceit (pride), mental stiffness, agitation, consciencelessness, and shamelessness are eliminated by the fourth knowledge.
  
  -- Vism. XXII.65 (Nyanamoli, trans)


The only case in which the inner monologue could possibly stop forever is at the moment of parinibbana, since that is the only event that could be said to be permanent. How else could one understand the word "forever"?

Even if one were to suddenly go without an inner monologue while still alive, there is no reason to infer that the monologue would not start up again; clearly both arahants and Buddhas still have some mental monologue as long as there is still the arising of sankharas.