It appears that he did not. On page 6 of the first issue of Watchmen, Rorschach appears in his civvies and is carrying his sign when he walks past the two police detectives that just finished looking over the Comedian's apartment. They are discussing masked vigilantes—specifically Rorschach himself—which piques his interest in the crime scene the detectives had just left.
On the next page, Rorschach finds the Comedian's badge—bloodstained, so Rorschach knows it is tied to the murder. It appears that finding the button is what convinces Rorschach that the crime definitely needs further investigation. He is still mulling over the fact that the apartment definitely belonged to the Comedian as he lays out the dead man's costume; "Hurm," he mutters.
The clincher, however, comes when Rorschach explains what he has found to his former partner.
Investigated a routine homicide. Victim named Edward Blake. Found the costume in Blake's wardrobe. Seems he was the Comedian.