Would it? If the laser is able to focus on the rocket for multiple seconds. Then it'd just need to pack a fraction of that energy over a sustained duration. Which sounds reasonable.
Indeed, most people are very surprised that a stick of dynamite has just the energy equivalent of a Snickers bar. Spread the energy release over more than just a few microseconds and you have a very different profile.
The rocket is moving at perhaps mach 4+. The laser has to strike a singular spot. So holding focus for multuple seconds is an as-yet-unsolved problem. Of course, hitting an incoming missile at all is at the edge of current capability.
I was interested in validating or invalidating the mach 4 number. I guess it depends on the specific rocket, but the 122 mm M-21OF-M Rocket, sometimes referred to as Katyusha (although that was the name of a specific launcher, not the rocket [0]) is 660 m/s or about mach 2 [1].
That's a good example of the problem. That's about 660 meters per second. Instead of "multiple seconds", the laser has about 1/100 second to penetrate the skin. Of a missile that's shaking, twisting, rocking.