Ice melting is simply the water molecules gaining enough kinetic energy (from collisions with the surrounding air molecules) that they break the bonds that held them in the ice crystal lattice. But at the microscopic level it's still just water molecules acting according to Newton's laws of motion (forgetting about quantum effects of course).
Now, back on the topic of the article: consider a system of 2 particles separated by some distance. Do they experience gravity? Of course they do. They start falling towards the midpoint between them. But where is entropy in this picture? How do you even define entropy for a system of 2 particles?