The Nurse's role in Medea is to bring the audience into the world of the play. In the prologue of the play, she gives the historical background, provides commentary on these events, and introduces the major characters: Medea and Jason.
At the end of the Nurse's opening monologue, the Attendant enters and tells the Nurse about the rumors being spread about Medea and Jason, providing additional background for the play.
The Chorus enters, and with the voice of Medea lamenting her current woeful situation in the background, the Nurse discusses Medea's tumultuous emotional state of mind and physical decline with the Chorus.
By the time the Nurse leaves the stage, never to return, she's served her purpose (her role), which is to give the audience the information they need to understand the rest of the play.
The audience never actually sees the Nurse in her role as "nurse" to Medea. The Nurse and Medea have no scenes together. The Nurse makes only passing reference to her "princess" and her "mistress," indicating that she holds a subservient position in Medea's household, but there are only a few lines spoken by the Attendant that give any hint as to her actual role:
ATTENDANT: Thou ancient treasure of my lady's room . . .
Will our mistress be
Content, this long time to be left by thee?
As for the Nurse never returning to the play, this is not at all unusual in Greek plays. Once a character like the Nurse serves their purpose, they were never seen or heard from again.
Also, until Sophocles introduced the innovation of adding a third actor to his plays, two actors played all of the speaking characters in a play, and only two characters spoke on stage at the same time. (The Chorus usually spoke in unison, "in one voice," as a single character, although the Leader of the Chorus often spoke to other characters.)
There are never more than two characters speaking on stage at the same time in Medea, and since the Nurse had already served her purpose in the prologue, the actor playing the Nurse could play other characters through the rest of the play.