An ICBM doesn't need to. An ICBM does not need to be that large. Being reliable and on demand is much more important than payload capacity. A Minuteman-III can launch with minutes of notice; you don't need to fool around with fueling it on the launchpad for a hour or more before launching it, or any of that nonsense. Because it requires less launch infrastructure, you can launch it from a greater range of locations. Being small also aids in this since it's a lot easier to move around the country. What's more, a Minuteman-III is $7 million a pop, which is a fraction of the going rate for a Falcon 9 launch.
So if your aim is to chuck nukes, a Minuteman-III is plainly superior to a Falcon 9 in every respect save payload. What about payload then? A Minuteman-III can carry up to three warheads with hundreds of kilotons of power each. That's a lot of damage. The Peacekeeper missile, removed from service in 2005, could deliver up to 10 warheads. Of course that cost more, a little more than a Falcon 9 launch, but still had the advantages of being a solid fuel rocket. Still in service, the submarine launched Trident-II can carry as many as 8-14 reentry vehicles, but in practice is limited by various treaties to a fraction of that. If there was the political willpower to violate those treaties and have an ICBM carrying more warheads than either the Minuteman-III or Trident-II currently carry, they would simply put more warheads onto the Trident-II. That'd be a lot simpler than re-purposing the Falcon 9, and a lot more useful.
Something else to consider though is range. Minuteman-III has enough range to hit Russia from the North, but not from the South. That might seem like a pretty severe limitation, but that's actually the way people want things to remain. Back in the 60s the Soviets designed something called the Fractional Orbital Bombardment System (FOBS). Basically, it was an ICBM capable of putting the warhead into orbit, giving it unlimited range. So, in essence, Russia could nuke America over the South Pole, instead of the North Pole, thereby bypassing all of the early warning systems that were looking North. Because this system was seen as a way of bypassing early warning systems, it was seen as a first strike weapon and therefore a destabilizing force. These sort of systems are now prohibited by treaty, and Falcon 9 would be in violation if it were an ICBM. But as a first strike weapon, a Falcon 9 is pretty shit. They take a long time to load and fuel, and their launch sites are high profile. A first strike is all about having the element of surprise; in an Falcon 9 FOBS/ICBM scenario, there are too many opportunities for detection. Particularly when you remember that SSBNs exist.
(There is another option though. The first ICBMs (as well as the Falcon 9) used cryogenic liquid propellants, and modern ICBMs use solid propellant, but in between these two technologies was a third: storable liquid propellants. The Titan II ICBM for instance burned aerozine and n204, both of which are liquids at room temperature, therefore the Titan II can be stored in a fueled state, ready to fly. These missiles could fly on very short notice, but they're also pretty dangerous to be around.)