Sigh. It is not that simple (it never is).
If by ‘white’ you mean an undyed nylon fibre, no. It would have no defenses. Undyed nylon is normally a slightly ‘dirty white’.
If by ‘white’ you mean a dye like TiO2 deeply embedded within the fibres, then yes, as the TiO2 particles would block the light from reaching the fibres. But I do not know of any such dye as a consumer item. And I do not know what can be used to make real ‘white’ either.
Light grey would (I think) suffer from a paucity of dye molecules.
In fact, it is my understanding that a carbon black ‘dye’ might be the most effective. Typically you can buy nylon stock (rods, sheet) in either raw (dirty white) or black. I find the black nylon to be stiffer than the raw – due I imagine to the carbon particles. I am talking here about nylon stock which is solid black, right through. (I use this for my winter stoves.) This is totally different from fabric.
However, that is not the end of the story. When dying fabric, the dye does not go in very far – maybe 20 microns, maybe more or less, it depends … An added complication is that when you search for ‘dying nylon’, what you will get will be half ads for fabric dyes like RIT. They are meant to be applied after the fabric has been made. Many of them are analine dyes, while others are extracts from various plants. All of those are ‘organic’, and that means they can be degraded by UV.
So how do factories dye nylon fabric? Usually with a dye bath, with high pressure and temperatures around 100 C. This has to be done in big batches, which is where we came in.
How do factories dye fabric in ‘camo’? With pad dyeing (ie daubing), which is not as effective, and still has to be done in large batches for economy and to give them time (distance along the fabric) to get the process running properly.
If you happened to live near a textile mill, I suspect you might be able to get some really good deals, albeit with erratic colours. But they are all in Asia.
Cheers