Can a jeweler remelt my gold ring into a new one?
Yes, that's possible. But it's trickier than it seems. Remember, a "gold" ring isn't pure gold. 14k gold is 58.5% gold, which makes it 41.5% something else. And jewelers use different combinations of metals for the "something else." So, unless the gold is melted down, the gold is separated, and it's mixed into that jeweler's standard alloy, it won't work.
Jewelers rarely cast one ring at a time - it's done in batches. So it's not as though they will melt OP's ring and reform it by itself. That's just not how it works.