He said that he did, but later in the episode River revealed that he actually said, "Look into my eyes," where she saw the real Doctor inside the robot. The name thing was a lie, as far as I know, so people wouldn't be suspicious about why he'd whispered to her, so he could pretend that he'd been killed.

But I guess it's possible he both told her his name and said to look in his eyes when he whispered her. But I think River doesn't know the name yet.