In the instructions it says the Diver can swim "across" a flooded or abyss tile for one action. Does this means he can make a turn and go to the tile diagonal from his location?
Yes, he can turn.
Page 4 of the rules state
The Diver may move through one or more adjacent missing and/or flooded tiles for 1 action.
There is a thread on BGG here where the designer confirms that turns are allowed.