The operating principle of proximity fuses is that they detonate the warhead in proximity of the target.
Some proximity fuses measure distance to target and can detect the moment of closest approach. Such method is used in torpedo proximity fuses and especially in naval mines, but is not usually needed or practical in missiles.
The proximity fuse of a a Sidewinder missile project narrow beams of laser light perpendicular to the flight of the missile. If any of the beams strikes the target it is reflected back to the missile where detectors sense it and detonate the warhead. There is no "measuring the range" at all, abd it is not needed - when a target is detected to the side of the missile then it's probably as close as it could be and is is good place to detonate the warhead.
Radar proximity fuses work on similar principle (beams of radio energy projected to the sides, perpendicular to misile or at an angle, to front-side) but they can have a predetermined maximum range at which the fuse can detonate. If the target is let's say closer than 17m the fuse would detonate warhead.
Usually there is no need to determine the "point of closesd approach" because the fuse anyway can detect a target ONLY to the side. So only when it passes by. And then in 95% of cases target is already as close as it can be (passing by) and the warhead (which is also designed to throw shrapnels to the sides) should be detonated NOW.
Radar fuse COULD measure range to target and detect optimal point for detonation, but this would be rarely usefull at all.