Overwatch Competitive Mode appears to have been delayed until mid-late July.
Responding to questions about the mode on Twitter, the official Overwatch account mentioned the new date:
@adrianmaturell We’ll be introducing our Competitive Play (hopefully) later in July. Stay tuned!
— Overwatch (@PlayOverwatch) May 27, 2016
This time frame only appears to apply to the PC version of the game. In a separate tweet, the same account said that “we don’t have anything to announce right now” in response to a question about Competitive Play on consoles.