I think what @Derek Lacy wrote is a large simplification although I'm no expert at all. To my knowledge water damage is covered - but not one caused by flooding (which, of course, is the cause here). Water damage caused by, for instance, a pipe bursting is - typically - covered (ask me how I know...). Also, what he lists is too limited as, again for example, theft or vandalism is - typically - covered.
It needs to be said, though, that renters insurance is typically a "named perils" type of insurance, meaning that the peril covered by the policy needs to be named/listed. If it's not named then it's not covered. Which, of course, then means that the matter of defining such perils comes into play: damage caused by flooding is not covered. But how do you typically get to flooding? The water doesn't just miraculously appear below the ceiling and fall to the floor causing flooding... But I guess if the cause for the flood was a pipe that burst then the damage ultimately caused by the flooding would be covered as the root cause of the water damage is a burst pipe. Those are my thoughts anyway.
@OP: you will have to take immediate remedial action as water damage can cause all sorts of secondary problems (mold etc.). You should inform your tenant about this and the necessary steps you envision to go with this (taken care by a professional). Also you should, at the same I guess, clearly communicate to the tenant that this was his fault and that he will be responsible for the damage caused. And no, it will not be left to his choice how this is going to be fixed. It's your building, and this is a serious incident. In addition you should check if the tub's overflow is indeed working (could have been clogged up by some kid having stuffed something down the holes or whatever, one never knows). Also, you should find out how this happened in the first place to avoid it happening again.
