May a player whose ball lies on a putting green brush aside casual water on his line of putt, or mop it up with a towel?
No. Such action would be a breach of Rule 16-1a (Touching Line of Putt). After reading the rule, you will realize that water is not considered to be a loose impediment.
16-1. General
a. Touching Line of Putt
The line of putt must not be touched except:
(i) the player may remove loose impediments, provided he does not press anything down;
(ii) the player may place the club in front of the ball when addressing it, provided he does not press anything down;
(iii) in measuring – Rule 18-6;
(iv) in lifting or replacing the ball – Rule 16-1b;
(v) in pressing down a ball-marker;
(vi) in repairing old hole plugs or ball marks on the putting green – Rule 16-1c; and
(vii) in removing movable obstructions – Rule 24-1.