Throughout the course of the novel, Nick is involved in several activities that would be considered immoral by many.
1. He breaks the law by drinking alcohol.
2. Jordan Baker implies that she and Nick were quite "involved" when she accuses him of using her at the end of the novel. This could mean that they had a physical relationship of some sorts.
3. Most significant is that Nick is privy to many activities and conversations that are immoral and does nothing to speak up. He knows of Tom's mistress and says nothing to Tom about his infidelity. He helps Gatsby reconnect and have an affair with Daisy, a married woman. And, he knows the truth about Daisy's being responsible for Myrtle's death, and does nothing to set the record straight.
Thus, while Nick is certainly a likable character--especially in comparison to Tom and Daisy--he is not particularly moral.