The voltage rating on a fuse tells you how high a voltage it can interrupt without arcing or otherwise allowing current to continue to flow after the fuse blows. A fuse of equal or higher voltage rating is fine as a replacement (after fixing whatever caused the old fuse to blow).
The conditions that will cause a fuse to blow are solely related to current (and time, for a slow-blow fuse). A replacement fuse needs to match current rating and slow-blow or not; no more and no less, so it doesn't blow spuriously and doesn't allow more current than desired without blowing.
Fuses are similar to a switch or relay that has to be able hold off a high voltage across itself. Interrupting a high current with an inductive load can create transient high voltages, which could create an arc. When a fuse first melts apart, you presumably do get some arcing over the tiny initial gap, but it presumably extinguishes quickly as more metal melts and the gap gets too long.