pull up the floor and replace rotten section and level. tile grout and seal new floor. use an approved expandable caulk around the tube/floor and there should be few problems. keeping the water above the tile is what you need to do, and the crack by the tub to the tile is most critical.
first thing to do is fix the leak, then rip up the tile and replace the section of floor that is rotten. If its just spash over, put down a towel when in the tub. a good tile job should be waterproof.
I'd go ahead..pull the tub...find where the water came from..put down a new sub floor..and floor..put a backer board up around tub...used to call this backer board green board..now they have one that's like sheet concrete(no, its not heavy....Lowe's has everything you need...not a bad repair for the do it your self-er
to be safe you need to replace the sub floor, then add lay down hardy backer board before you tile the floor.