Similarity with the linked question
The scenario in your case is essentially similar to the one in the question you linked. How? Because once the ice cube starts melting, the water (molten ice) forms a layer beneath the ice. So now the ice is no longer in contact with the table, but it's instead in contact with the thin layer of water separating the ice cube and the table. This layer of water is clearly visible in the video that you linked to. This ice-on-water scenario looks something like this:
(source: gstatic.com)
Image source
So, as you'd have anticipated, the physics isn't going to change much (though there is a significant difference in this scenario, see the rest of the answer for more).
Explanation
Temperature dependence of density
The reason behind the rotating ice cube is fundamentally because of the variation in density of water with temperature. The density versus temperature graph for water looks like this:
Image source
As you can see, around the temperature range from $0^{\circ} \:\rm C$ to $4^{\circ}\:\rm C$, the density of water increases with increase in temperature, but after the $4^{\circ}\:\rm C$ mark, it continuously decreases with increase in temperature.
Difference between the your question and the linked question
Now, let me introduce the difference between both the scenarios that I was talking about. In the other question, the OP has put ice in a container filled with hot water. This directly implies that the surrounding hot water has lower density than water at any temperature lower than that of the hot water. So the idea of creation of vortices is easily explainable. However, in your case, the surrounding water is not at all hot, and thus the density increases with the increase in temperature, so we might not be able to extend the same explanation here as well.
Then why are vortices formed?
Let's use some approximate temperature data to know why. Let's assume that the table on which you're putting the ice is at a temperature of about $30^{\circ} \:\rm C$ and the ice is at temperature $0^{\circ}\: \rm C$. Thus the water surrounding the ice will also be at a temperature of $0^{\circ}\: \rm C$. And the water touching the table will be at a temperature somewhat around $15^{\circ}\: \rm C$. Now, from the graph of the density versus temperature, we can clearly see that water at $15^{\circ}\: \rm C$ has lower density than water at $0^{\circ}\: \rm C$. This is because the maxima of the approximately parabolic density-temperature graph occurs at $4^{\circ}\: \rm C$ which is more closer to $0^{\circ}\: \rm C$ than to $15^{\circ}\: \rm C$.
So, in your case, the water surrounding the ice (at a temperature of $0^{\circ}\: \rm C$) is heavier than water at, say, about $10 ^{\circ}\: \rm C$ or $15^{\circ}\: \rm C$. Thus it would get replaced by the water beneath the ice (the water touching the table). In other words, the water surrounding the ice sinks, pulling inside the water from the periphery (whose density is more than the water at $0^{\circ}\: \rm C$). And, voila, there are our vortices.
Vortices
These vortices interact with the floating ice cube and thus the floating ice cube, initially stationary, starts beginning to rotate.
Is the conservation of angular momentum violated here?
No. The conservation of angular momentum is definitely not violated. The angular momentum of the generated water vortex is exactly equal and opposite to the angular momentum of the rotating ice cube. Slowly, the angular momentum of this vortex is transferred to the table beneath the ice cube.
Direction of rotation
The direction is as such ambiguous (and difficult) to decide in this case, but in the generic case of an ice cube in hot wwter, the direction primarily depends on the inital state of the water inside the container. If initially the water is rotating in a certain direction, then the ice cube will rotate in that direction. In this case, though, since there is no initial rotational flow already in place, thus other factors like the shape of the ice cube start dominating the direction of rotation.
Further experimentation
To validate or invalidate the explanation I gave above, you can conduct the following variations of the same experiment.
Removing the water beneath: First, do the original experiment as usual. Then once the ice cube starts rotating, lift the ice cube and wipe out all the water beneath it and place it on the platform again (alternatively you could also lift the ice cube abd change the ice cube's location to a dry place). Now again observe the ice cube. According to my explanation, the ice cube shouldn't rotate initially and stay stationary, until a layer of water develops beneath it. Check if this holds true.
Changing the temperature of the platform: Now, redo the experiment, but choose a platform whose temperature is in the $1^{\circ}\: \rm C\: -\: 7^{\circ}\: \rm C$ range. In this situation, the water which is not around the ice cube should be at a temperature nearer to $4^{\circ}\: \rm C$ and thus it should be denser than the water surrounding the ice. This implies that the water around the ice does not get replaced and there is no vortex formation. Thus, according to my explanation, the ice shouldn't rotate in such a case.
Also, there might be a possibility that the ice cube is rotating here just because it might have been given some initial angular momentum, so it just keeps on rotating with almost the same angular velocity due to the negligible resistance provided by the water beneath it. But this is very less likely since there is no appreciable decrease in angular velocity in the video that the OP linked to (in fact the angular velocity increases at some points), so it's not quite likely that all this rotational motion is caused just due to the inital angular momentum imparted to the ice cube.
You can try out the above experiments to verify the explanation. And if anything goes wrong, please do inform me so that I can correct my explanation :-)
