No. The text never says that he cut himself witht the razor. He felt the talisman and got out of the water.
Suicide, both committing and imagining, is a recurring idea in Steinbecks novels. The "best" one the slipping of Kate Trask into Alice's shoes just before taking the poison. And than vanishing down the rabbit hole. In the original manuscript Steinbeck left it open whether Ethan Hawley died, only to add some more lines upon the request of his editor.