Now, one good answer to "could deliberately induced Kessler Syndrome stop ICBMs?" is that the amount of mass required for such a thing is in the realm of ~2.5 trillion kilograms of matter, which is multiple orders of magnitude larger then the total amount of mass humanity has put into orbit.
However, what if it wasn't humanity doing it?
Let's say that a self-replicating pseudo-Berserker probe enters the Sol system. Its objective is not to exterminate all life on Earth, or to terraform it, or to render it uninhabitable; instead (hence the "pseudo-" prefix), its modus operandi is to:
Create a series of local system-only tugboats and gravity tractor craft that redirect asteroids into the lowest possible stable orbit of worlds it deems to be habitable.
Mine these asteroids apart in order to self-replicate, refuel (after all, hydrogen matter-antimatter annihilation rockets require hydrogen as well as antihydrogen), and build more self-replicating probes, as well as several kilostructures (think of them as smaller versions of a megastructure) in the outer Solar System. The exact number of self-replicating probes and kilostructures built depends on the plot.
Disperse leftover/unutilized materials into a relatively dense orbital shell which covers the entire planet, so as to make it impossible for spaceflight or space travel to occur until laser broom technology is invented. This is a deliberate design choice by the probe's creators (ancient humans); they believed that one of many metrics by which a responsible civilization can be distinguished from an irresponsible one is their level of interest in developing and using anti-orbital debris technology. This has a side effect of stopping ICBMs, which are exposed to flying particles during their midcourse phase and are hit with high-velocity rocks and gravel, damaging them and rendering them incapable of delivering their payload.
Boost out of the system to construct more kilostructures and probes in other ones. Alpha Centauri is next.
In short, mass is really not a problem, because, from a writer's viewpoint, this thing can use as many asteroids as it needs to self-replicate, and its secondary goal is to make any orbital altitude inside a geosynchronous orbit impassable, at any inclination.
The question: with mass removed as a constraint, now is it possible for deliberately induced Kessler Syndrome to stop ICBMs, at least until laser broom technology is invented and rendered operational?