Do the recombination of charge carriers cause the diode to stop conducting?
As the injected minority carriers recombine, they have to recombine with something. What they recombine with is some of the majority carriers. So majority carriers near the junction are being "used up" by the recombination process.

As the majority carriers are taken up by recombination, new majority carriers flow in from the area farther away from the junction. This majority carrier flow is what carries the current in the "far away" part of the device.

I found a diagram showing this here: