Romeo does, in fact, leave Verona because he stays in Mantua per the instructions of Friar Laurence. For, it is Mantua that is quarantined because of a plague, preventing Friar John from giving Romeo the note that informs him of the scheme of Friar Laurence and the news that Juliet is, indeed, alive; it is only a drug-induced sleep in which she rests.
Romeo's cowardice, then, is in his act of suicide. For, he is too afraid to live without Juliet's love. After he is informed by Balthasar that Juliet lies in the Capulet tomb, Romeo is ready to commit suicide and join her in death, too frightened to live his life alone. Then, once he flees Mantua and enters Juliet's tomb, Romeo tells Paris who has also entered,
Oh, be gone.
,,,,I come armored against myself (5.3)
When he sees Juliet, he kisses her, takes the potion and dies, avoiding living alone..
Romeo is a hero because he did not back down from Tybalt and saved his friend, but he is a coward because he refused to leave Verona.
Romeo is a hero because he stood up to Tybalt in the fight, which proves that he is not a coward in a fight. He was standing up for his friend Mercutio. When Tybalt was there with his sword, and Mercutio was in danger, Romeo did not back down. Of course, he might also have done more harm than good, at least according to Mercutio (he claims to have been hurt because of Romeo).
Mercutio dies, and curses both Romeo and Tybalt. When Tybalt returns, Romeo does not hesitate to come after him.
Now, Tybalt, take the ‘villain’ back again
That late thou gavest me; for Mercutio's soul
Is but a little way above our heads,
Staying for thine to keep him company.
Either thou or I, or both, must go with him. (Act 3, Scene 1)
He must have been a little scared, but he was probably angrier than scared. He was able to defeat Tybalt, and in fact kill him—even though this was his love’s cousin! At the time, he did not have much choice. Tybalt came back, and someone had to fight him.
When it came to facing the consequences of his actions, Romeo was not as brave. In fact, he turned into a big baby. This is why I would say that he was a coward when it came to his banishment. He seems to respond to his banishment by throwing a tantrum.
O holy friar, O, tell me, holy friar,
Where is my lady's lord, where's Romeo?
There on the ground, with his own tears made drunk. (Act 3, Scene 3)
So although Romeo faced down the fearful Tybalt and defeated him in battle, when faced with banishment he is on the floor bawling his eyes out. Yes, it’s true that he has to leave the city, but is it really that bad? Can’t the two of them just go? Grow up already! Romeo is being silly and cowardly. If Juliet means that much to him, he will find a way to make it work.
Alas, the solution the two of them come up with does not work out well for anyone, and as we all know it ends in tragedy. Romeo might have had some strength, but it might have been false bravado. When it came down to it, he did not have much real bravery. He was young, naive, and immature. We can't all be brave all the time.
I agree that suicide was cowardly, but I was referring more to how he reacted to being banished. That is why I included the quote about him crying and being on the floor. While that might be the nurse's opinion, if it is an accurate description of his behavior it matches how he reacted to the news.
There is no world without Verona walls,
But purgatory, torture, hell itself.
Hence banished is banish'd from the world,
And world's exile is death. Then ‘banishment’
Is death misterm'd. (Act 3, Scene 3)
Thus, the suicide is an extension of his reaction here. He is not taking responsibility. He is a husband now. He needs to be thinking of someone other than himself. Yet he is only considering how this affects him. Yes, he follows directions from Friar Lawrence. He is a child, but he is still acting like a child. That is why I called it a tantrum.
He wanted to be a man. He wanted to either protect his friend or avenge his death, depending on your point of view. So why is he now behaving like a baby? Now is not the time to whine and throw a fit. Now is not the time to blindly follow orders. Now is the time to, perhaps, take advice. However, he should be thinking of Juliet and taking control of his own life and hers and planning for a future together for them. Instead, what he does is an act of cowardice. He throws a fit, and then he goes with his tail between his legs. Coming back, committing suicide when he thinks she is dead, all of those are further extensions of the same pitiful acts of cowardice. He cannot face the consequences of his actions. He cannot be a hero then, as I said before. That one act of heroism was nothing but hot air.