The Food and Drug Administration approved a controversial drug for depression Tuesday that could be the first of a long-awaited wave of new treatments, but has also raised concerns about abuse.

The drug, a nasal spray that manufacturer Johnson & Johnson has branded Spravato, is a close chemical relation to ketamine, an anesthetic that is known for its heady effects on the brain. It is often abused as a party drug with the nickname “Special K,” but it has also been shown in studies to have a fast-acting impact on depression...