No, horses generally don't wear socks under their shoes.
Socks are white markings that start at the hoof and extend to some point between the fetlock and the knee or hock.
A horse might have socks on one leg all the way to all four. Note that in some equestrian disciplines, horses with a sock on only one side are frowned upon because it can make their gait look uneven.
The chestnut horse in the center has a sock on his right hind leg. And is hamming it for the camera. Or asking me to take all that gear off his back, not sure which.