I think the movie is deliberately ambiguous at the end to make you internalize an examination of your own personal integrity. However, exploring the plot leads one more to think Stephen did go public with the truth.
On one hand Stephen says that a candidate can start all the wars they want and bomb anywhere they want, but they can't have sex with their intern. So he doesn't consider what Morris did as that bad, until Molly Sterns commits suicide over Zara firing Stephens, presumably because she thinks Zara found out Morris getting her pregnant and Stephen having sex with her. It's what a strict religious girl might do, especially when she's about to have an abortion, which she doesn't believe in either. When Morris isn't upset over Molly's death enough for Stephen, Stephen is sufficiently upset enough, along with getting fired by Zara who blows Stephen's meeting with Duffy all out of proportion, as an excuse to fire him, because he knows Stephen is better than him, turns Stephen against Zara and against Morris who antagonizes him when they meet to talk about Stephen exposing him publicly if he didn't get Zara's job.
Molly's death and then Morris glossing it over and then Stephen getting fired was enough for a man with principle and integrity, as Stephen was, to sacrifice himself and ruin Morris's campaign.