As I suggest in my ACOK chapter essays, I think Melisandre told him that it was a necessary ritual (to prove his faith or to purify him or something), but didn’t tell him why it was necessary or what it would do. After all, “Melisandre has gazed into the flames, and seen him dead.” If R’hllor is the one true god and Melisandre’s visions are true – both propositions that Melisandre has gone to great lengths to try to convince Stannis of – then there’s no need to take action, much less to use black magic (which would seriously undermine those propositions) to kill him.
And after the fact, I think Stannis was in deep denial about what he had subconsciously experienced in the night – “I dream of it sometimes. Of Renly’s dying. A green tent, candles, a woman screaming. And blood….I was still abed when he died. Your Devan will tell you. He tried to wake me…Devan says I thrashed and cried out, but what does it matter? It was a dream. I was in my tent when Renly died, and when I woke my hands were clean.”