Verified answer

 You most certainly could. I would caution you to make certain you would NOT be purchasing the 3GB model as this will not give any real overall performance boost. I would caution against buying the 1060 in general due to the more recent and reasonably good value 1660 which also comes with more RAM resulting in a very good mid level card.

As to PSU, you shouldn't need more than you have right now for either card. I have an RX570 4GB and it idles using 35W and under full load when doing video transcoding it uses about 75W or so. The temps which are IMHO a bit more critical show this card never goes above 75C. Your system should be fine with the 460W especially if there is no over clocking.

Nvidia recommend a 430w Power supply for either version of the GTX 1060. Furthermore, when gaming the GTX 1060 uses less power than an RX 570. Your 460w power supply will work. Another card that a 460w power supply can handle is the GTX 1660ti or 1660 Super, which would 

The GTX 1060 6gb version isn't much faster than an RX 570.... maybe by 12% to 15% if the card has a factory overclock. So if you're getting 40 fps with the RX 570, then you can expect 46 with the 6gb GTX 1060. That's not a justifiable upgrade. To make matters worse, the 3gb GTX 1060 is slower and it's damn near on par with the RX 570. 

The GTX 1660 Super or the GTX 1660ti is around 50% faster. Another card that would be a great upgrade if you can afford it is the 6gb RTX 2060. The RTX 2060 is stretching it but it would work with a 460w power supply. 

it is best to use a psu sizing app. they are found online -- search for one.

warning: many cheap psus do not actually deliver the rated power -- or don't deliver it for very many years

Is it compatible with the motherboard?