"Stone" isn't really a thing, in the way think of it. It's not a simple substance like plastic, glass, or metals that can be melted and molded.
The properties that make a stone come from a complex interplay of many factors. Stones made from exactly the same elements and molecules can behave in radically different ways depending on how quickly or slowly they cool.
Marble, specifically, is composed of calcium carbonate crystals. Normally it has started out as limestone, produced by biological processes, which is subjected to heat and pressure that causes the calcium carbonate to crystallise in an interlocking pattern. If you were to take that marble and melt it at atmospheric pressure, you'd actually destroy the calcium carbonate, and be left with calcium oxide and carbon dioxide.
To melt it down while maintaining its chemical composition, you'd have to subject it to enormous pressure while heating it to around 180°C at the same time. This wouldn't melt the marble, but it would allow you to reshape it. You would then have to control its cooling very precisely, to allow it the time to form the interlocking crystal structure that makes it marble. This would likely take a long time, though I can't find exactly how long; generally, the more slowly something cools, the larger crystals it would form.
The same answer applies to most other rocks as well. In most cases, if you simply melt down a rock and let it cool, you're going to get some form of glass. After all, we make glass by melting sand, and sand is simply pulverised rock.
The way minerals cool is actually more important to what they become than what they started as. It's all about the size of the crystals you can form in a given sample.
Take a look at this:
This is a piece of nickle/iron alloy. Just iron, the same as your car is made of - but you'll never be able to make a piece of iron that shows a pattern like this. It's called a Widmanstätten pattern, and it will only form when the iron cools slowly. Very slowly. Like, over millions of years-slowly. It takes that long for large iron crystals to form.
Marble, specifically, is similar - it needs to cool over thousands of years for its crystals to form their interlocking structure. Granite is composed of rocks that have been melted and then allowed to cool over thousands or millions of years. Sedimentary rocks that are melted can cool into granite. Rapid cooling of the same liquid rock would leave you with dense basalt or obsidian.
So, while it's not theoretically impossible to use a combination of heat and pressure to reshape a sculpture or a carving back into 'virgin' stone, and while we could possibly do it today if we had a strong need, it's nearly always going to be much more practical to simply dig a new block out of the ground.