Think I'd buy him a set of soft toenail covers from Amazon, let him wear them for a week elsewhere in the house where he's comfortable and then carry him into the room he currently fears, put him down and see how he does once he realizes he's no longer slipping. But then I don't know much about dogs.
We've always called it the linoleum gene.
Some breeds have the fear more that others, but any can develop it.
Both Belgian Sheepdogs i've had were like that. Aussie and the mutt didn't care one bit no matter how many times they wiped out around the fridge or the hallway corner.
But the Belgians - terrified of it. Wife always has runners in the kitchen for him (the current one) and couple strategic small rugs. There is no laid carpet in the house. The previous one would NOT go on the hardwood at all unless there were runners to hop to. The current one will go through the house, but he does it with trepidation.