The floor is definitely stone (travertine). The walls are either travertine or porcelain, but most likely travertine. Also, are you sure that's water deposits and not soap scum? It's hard to tell from the pictures.
When it's this bad, I use acid to make the cleaning faster, then re-hone wherever it's affected. The honing adds more time, but with the time you save by using acid, it's pretty much a wash. Just don't use straight muriatic or anything. You can get it off in layers. Let the acid fiz, scrub, re-apply, fiz, scrub etc.
If it's soap scum (waxy feel, as apposed to rough feel) MB-3 soap scum remover is the way to go. It won't affect the stone.