The gemara says that one can't crush ice in order to turn it into water. Rashi explains that the issur is that they are creating something new which is an issur d'rabonon since it resembles a melacha. But one may put it into a cup or plate, which rashi explains is referring to allowing ice to melt in a cup of wine. Rashi's approach to this gemara seems problematic because it begins with the assumption that there is a ma'aseh issur in crushing the ice, but then implies that if they would allow the ice to melt in an empty cup the resulting water would be assur even though no ma'aseh issur was done.
The Ran offers two approaches in explaining this gemara. The first approach is based on Rashi and Tosafos (not found in our tosafos) that the nature of the issur is a ma'aseh issur, therefore it is only forbidden to crush the ice but permitted to let the ice melt by itself. According to this approach there is no issur in the result or product, only in the act of crushing the ice to change its form. The Ran citing the Rambam follows the same approach, just that he considers the issur to be a subset of sechita, squeezing fruits that are meant for their liquid, rather than an issur of creating something new which isn't associated with any particular melacha. The Ran then offers an alternate approach in the name of the sefer hateruma who seems to understand that the ma'aseh isn't the problem, rather the result is the problem. Ice that transformed into water is muktzah since it is nolad on shabbos and didn't exist in that state before. According to this approach there is also a prohibition to do an act that changes the form, but the nature of the prohibition is the product or result, therefore one cannot do an act to yield that result. Based on this second approach it is prohibited to use the water that came from melting ice, and one cannot do something causing ice to melt or congealed fat to melt into a liquid form, such as putting it near heat or in the sun. The only heter in the gemara is when the ice melts into liquid that is already present so that the nolad product is never realized on its own.
The Shulchan Aruch (320) seems to follow the first approach and therefore only forbids active crushing of ice for the purpose of making water, but permits one to allow ice to melt. However, the implication of the shulchan aruch is that one can only use the melted ice when it melts into water or wine, which is problematic because if the S.A. follows the first approach the product should be permitted. Furthermore, the Shulchan Aruch in 318 clearly permits one to place congealed fat near a heat source in a way that it will melt implying both that it doesn't qualify as actively melting it and implying that the product is always permitted even if it isn't mixed into something else. Therefore, the Shulchan Aruch in 320 that permits ice that melts into wine or water, would hold that even ice that melts into an empty cup would be permitted following the first approach, but chooses a more common case such as melting ice into a drink rather than into an empty cup. This would also seem to be the approach of Rashi that the nature of the issur is the ma'aseh issur, not the resulting product. Rashi explains that one can melt ice into a cup referring to a cup that contains water or wine simply because that is the more common case. This seems clear from the fact that Rashi says בימות החמה, he seems to be focusing on the fact that it is summer and therefore the wine wouldn't taste good unless it is cooled down with some ice. But really both rashi and the shulchan aruch would permit one to place ice in an empty cup, put it near a heat source and drink the water.
The Rama (318) on the other hand writes ויש מחמירין implying that one should be machmir like the sefer hateruma. Even if there is gravy into which the congealed fat will melt, the Rama holds that one must be machmir since the melted fat will sit on top of the liquid gravy and be considered nolad which is muktzah. According to this approach any frozen or congealed item that changed form and is recognizable as an entity without being mixed into something would be forbidden to use and since it's forbidden to use it is also forbidden to make. Therefore, one cannot use the water that results from ice melting in an empty cup.
The opinion of the sefer hateruma who considers the problem to be in the product and not the action would seem to result in a leniency. Is one allowed to actively crush ice causing it to melt in a cup of wine? The first approach (shulchan aruch) would consider this to be an act of molid or sechita and therefore be assur, but the second approach (rama) would hold that since the result is immediately mixed in with the wine, it isn't a nolad problem and therefore permitted. However, it seems that the Rama is machmir for the sefer hateruma, but doesn't pasken like the sefer hateruma against the first approach, and would therefore admit that it is assur.
It would also seem that according to the sefer hateruma that the rama is machmir for one would not be able to make ice on shabbos either. Since the nature of the issur is the product being muktzah, there should be no difference between the melting of ice or the freezing of water. In both situations the product should be muktzah and therefore assur to cause.