Well, I have a theory about how this scene might have been filmed, but I have no sources to confirm this as it is a very old movie.
This rock can be a fake one, because using a real one will definitely hurt an actor.
Now, when the rock is thrown, actor takes a jump.
To make this look realistic, the timing and jump must be perfect.
If you take a close look, you can see that Charlie Chaplin just waited for the perfect time and made a jump (credits to @DrRDizzle we can see Charlie Chaplin's knees bending slightly.).
Such scene might have taken practices and takes for a perfect shot.
movies.stackexchange.com/users/27264/A J
In addition to the fake rock theory by @AJ, it is quite possible that the rock did not travel the entire distance it appeared to.
Once the rock disappears off screen, there could have been some barrier to prevent the rock continuing.
This way, the rope would have enough weight to continue moving, but there would be significantly less weight pulling on Charlie Chaplin's neck.
movies.stackexchange.com/users/41339/Matt Butler
Even if this particular trick is rare, it's conceptually similar to hanging scenes, which are abundant in westerns.
In fake hanging, the typical trick is to attach the rope to the hidden harness while the knot is just a decoration:
How is a hanging scene filmed?
In addition, the "stone" is likely to be light.
In our days, I'd say plastic.
Then, probably papier-mâché.
movies.stackexchange.com/users/22809/IMil
The key is to watch Chaplin's legs and the rope.
As the rock goes off-screen and the rope is getting close to taut, his knees flex.
At that point he clearly jumps into the water, with enough style (because he was an expert stuntman) to look as if he'd been pulled into the water by the rock.
If you watch the rope too, you see that there's no point at which it goes taut, and actually when he jumps into the water it goes completely slack.
This is completely clear when you watch it back repeatedly and analyse exactly how it's done.
For audiences though it happens fast enough that they don't see the knee-bend and jump, or think through the implications of the rope going slack.
As a result they buy into the scene, and no doubt some of the original audience asked the same question you just have!
You can also be sure that the "rock" is nothing of the sort.
For the stunt to be safe in water, the "rock" will almost certainly be balsa wood or something equally light, so that it doesn't genuinely sink afterwards!
Notice that the "rock" goes off-screen to ensure you don't think about that - and
notice that whilst Chaplin produces a very impressive splash from his pratfall, there's no splash from the rock, even though the rope could only let it go just off-screen.
movies.stackexchange.com/users/30415/Graham
Perhaps the rope is loosely fastened to the rock, and detaches when the rock has reached its limit.
movies.stackexchange.com/users/41350/Pete
I've watched it several times
and I think: the rock is firmly attached to rope, and the rope to Chaplin, albeit to his shoulder not his neck.
if you watch carefully, the rock reached the extent of the rope and was bouncing back -- you can see the rope go taut and then slacken.
the "rock" was made of some moderately light-weight material, most probably papier-mâché as soon as Chaplin felt the rope tauten, he hopped convincingly off the dock (the man was, after all, one of the great physical comedians of all time).
movies.stackexchange.com/users/9071/Malvolio
It's a cut.
It's two takes spliced together to make it seem like a single action.
Closely looking at the gif, i.e. the video, shows where there is a significant difference between the frames.
Frame 38 Frame 39
Every other frame is a subtle change from the last.
This one shows a significant difference.
Charlie's face jumps to a different position.
The other guy's hands do too.
The Rope seems to have been matted in.
The rock suddenly disappears.
Primitive Movie Magic at work.
movies.stackexchange.com/users/23541/cde
Try lifting, let alone throwing a rock of that size and you'll understand how that was fake; it has just the right weight to pull the rope in a realistic way, and then Chaplin only has to jump at the right time.
In fact, notice that he does a little funny jump, not
a "rock&rope pull my neck suddenly and nearly breaks it" one, which would be the more real way.
Sorry, this is very similar to the top answer.
movies.stackexchange.com/users/41359/JustPassingBy
The rock could be fake and there could have been a hidden rope tied to a body harness on Chaplin that could have done the pulling from off-camera.
I'm not convinced
his posture and movements give enough evidence to prove conclusively that he jumped.
I'm not seeing enough upward force from his legs to justify the height and length of the jump.
Hence, I suspect a hidden rope pulling him from the side, off camera.
However, being pulled from the side by a hidden wire could easily explain the movements.
movies.stackexchange.com/users/8944/Thom Blair III
In a scene from  City Lights  (1931), a man tries to commit suicide before Charlie Chaplin comes and saves him.  But the man accidentally ties a rope to Chaplin's neck and throws the rock into the water with some force.   How was this scene shot safely without breaking Charlie Chaplin's neck or injuring him in some other way?
How was this scene shot without hurting Charlie Chaplin?
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<film-techniques><stunts><city-lights>
movies.stackexchange.com/questions/60924
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The Charlie Chaplin scene could have been filme like a fake hanging. However, the key is to watch Chaplin's legs and the rope. As the rock goes off-screen and the rope is getting close to taut, his knees flex. At that point he clearly  jumps  into the water, with enough style (because he was an expert stuntman) to look as if he'd been pulled into the water by the rock. If you watch the rope too, you see that there's no point at which it goes taut, and actually when he jumps into the water it goes completely slack. This is completely clear when you watch it back repeatedly and analyse exactly how it's done. For audiences though it happens fast enough that they don't see the knee-bend and jump, or think through the implications of the rope going slack. Hanging scene is fake hanging. 
Hanging scenes are abundant in westerns. In fake hanging, the typical trick is to attach the rope to the hidden harness while the knot is just a decoration. The "stone" is likely to be light. In our days, I'd say plastic. The key is to watch Chaplin's legs and the rope as the rock goes off-screen. At that point he clearly jumps into the water, with enough style to look as if he'd been pulled. If you watch the rope too, you see that there's no point at which it goes taut.