All that would happen is you would make a hole in it, but since it's on a ballistic it would keep on going. (Ballistic means gravity is controlling it, it's not powered.)
Even if you blew it up, at best you would spread nuclear material all over the target area.
You need to hit these things during the boost phase, where a: you can stop it from going ballistic and make it fall back into the launching country, and b: you have something volatile to blow up (the fuel).
Also, it's easier. When it's at the apex it's moving horizontally very quickly, and it's at it's closest to you. Which is the most difficulty when it comes to aiming (since the angle of the shot is moving the fastest at that point).
When it's moving vertically toward you is much easier to aim, since you barely have to move the laser.
A high-powered laser won't just heat stuff up. If you can pump enough energy to small enough surface area in small enough time, the outer layer of the object being hit will be ionized. The laser will pump even more energy to this cloud of electrons and ions. For all intents and purposes, this cloud can be called an explosion.
Yes, I'm saying that when you hit a piece of garden-variety steel with a strong-enough laser, it will blow up and disintegrate into small pieces.
And even if the missile is moving very fast horizontally at the apex the movement is entirely predictable, even minutes into the future. All you need is two rangings, then you can just point the laser at where the missile is going to pass trough.
And anyway, you can predict the motion as soon as the motor is finished, you don't have to wait for the apex.
It's very difficult to aim the laser that well - even if you know exactly where it's going to be. Even if you were perfect in location and control, random atmospheric effects would mess it up.
The only practical way is via feedback loop, and when the angle is changing so fast it's very very hard to do.
You want a place where the angle doesn't change much, and all you need to do is adjust it based on feedback.
As far as destroying the missile - you need to pump enough energy into it to vaporise the entire thing. Normally an explosion will fracture the object near it. That cloud won't. You are moving very fast in air (even at the apex), and any ions will be blown away, plus they are only moving away from the missile, and not toward it, so they will not damage it. (They are moving away because they came from it, when they vaporized they bounced off the surface and away from the missile.)
You have to vaporise the entire thing layer by layer. And the energy in the cloud will not help you - since it's moving away from the missile it won't heat it, in fact it will insulate it (until it's blown away by the air) because all the energy of the laser will heat the cloud, and do nothing to the missile.
It's too easy to defend against a laser anyway. A small amount of water in the nose, released when it detects a laser will produce a cloud of water that will absorb all the laser's energy.
Not in a space based laser aiming at an ICBM at it's apex, it would be about 500 to 700 km depending on the trajectory, and doing about 14,000 km / h.
Unfortunately the people that launch these things are well aware of that, which is why if they are launched they are not launched 'one-by-one' to allow you to recharge your lasers between shots.
Another slight problem with the strategy is that any attacker capable of launching an ICBM at a country would probably also have the capability to hit that platform, and wich a 1GW radiated output it provides you with a really nice fat target.