I think you might run into an issue here. Now with the word 'apocalyptic', that's pretty much guaranteed, but with the word 'clap'.
Contrary to popular belief, when you clap you aren't simply throwing your hands together. Your hands, arm muscles, brain and even eyes are engaged in complex interplay to make sure your hands collide just so (if you want proof your eyes are involved try placing just the tips of your index fingers together with your eyes closed). Your hero has infinite speed, which removes the need to worry about nerve signal transmission time (except it doesn't, but I'll get to that later), but what he doesn't have is a nervous system designed to deal with relativistic lag.
It's well known that when you reach relativistic speeds weird things start happening with time, notably you have to start worrying about how old your twin is. Our hero has the problem of his brain having to worry about how old his hands are. For a bit of reference: we have to correct for relativistic lag in satellite communications.
The amount of time it takes a signal to reach a satellite at the top of the Low Earth Orbit boundary is about 5ms. The amount of time it takes a normal human nerve signal to get to the end of the arm is about 5ms (thank you, XKCD). The satellite is moving at a speed considerably lower than c. Unless our hero can perform relativistic reference frame substitution in his head
while tied to a centrifuge then he's going to run into issues with nerve signal interpretation.
Now, I don't claim to know what happens when a brain is sending messages to muscles that are sending feedback at a completely different rate, and I wouldn't even want to try modelling a neurochemical reaction on a changing temporal boundary. If you speed up the signal transmission you just end up with more confusion, as the problem isn't with speed of signal but differences in the rates with which the signals are being sent and intercepted. If you say that the nerve signals are instantaneous then you end up with a causal violation. From the point of view of the brain the clap happens before it's sent the signal for the clap to begin.
With this amount of bodily confusion I reckon that our hero isn't going to have the level of fine motor control required for a good, solid 'CLAP'. In fact, I'd wager that there would be something more like a localised seizure in their biceps, followed by the most apocalyptic derp in history.
Even if their hands don't collide at all (like missing your own high-five), our indestructible hero's hands will stop in some way, most likely a highly uncoordinated pirouette. Assuming that by 'indestructible' you mean they are in no way affected by external forces and that nothing can enter their body their arms will leave a trail of vacuum where they've pushed all of the air ahead of them. The leading edges of their hands and forearms will be a thin incandescent layer of plasma where adiabatic compression has taken place, and the sudden shock of their asymmetrical epic flail will briefly transmit an ungodly amount of energy into the ground below their feet (This wouldn't happen if their actions were perfectly symmetrical, but as they aren't their feet will be exerting a nontrivial torque on the floor, much like if you swing one arm while on a swivel chair)
The exact amount of energy transmitted to the floor will be somewhat under ckersch's excellently calculated energies, but as you can see from the ungodly exponents that he produced, even if our hero's indestructible shoulder sockets re-absorb 99% of the energy involved we're still talking continent-crackingly huge numbers. Oh, and the air around our hero is now a combination of on fire and propagating a huge shockwave from the vacuum implosion. And that's before we figure out if our hero's shoes and shirt have vaporised from the friction.
All in all, maybe not an apocalyptic clap, but it's certainly apocalyptic. And you can guarantee his immortal buddy will somehow get a photo of it and put it on whatever's left of instagram with a caption like:
'Hurr-Durr, Imma destroyin' the wurld!'.