Is it correct to say that if the P-N junction is in the middle of the depletion region, half of the depletion region is for the P region and the other half is for the N region?
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Not true. The depletion region width depends on the doping. You can think that the non-moving +charge ("the bound charge") in the depleted N side has as much charge as the depleted area in the P side has non-moving minus charge. Lower doping means wider depletion zone and there can be very different dopings in the N and P halves.
The same is explained with some equations here https://en.wikipedia.org/wiki/P–n_junction
