Before determining whether Romeo is a tragic hero or not, it's important to understand the characteristics that tragic heroes must encompass. I provided a reference link at the bottom of this answer to a different eNotes thread on the topic of what makes a tragic hero - check it out for more detailed information. In summary, a tragic hero must:
- Be born into nobility
- Have a tragic flaw (also known as hamartia) - a flaw that leads to a character's downfall
- Be fated to make an error in judgement
- Evoke a sense of pity from the audience
- Be responsible for his/her own fate
- Lose nobility or status after the error in judgement
- Experience a tragic death
The prologue of Romeo and Juliet give us many clues as to whether or not Romeo could be considered a tragic hero:
Two households, both alike in dignity,
In fair Verona, where we lay our scene,
From ancient grudge break to new mutiny,
Where civil blood makes civil hands unclean.
From forth the fatal loins of these two foes
A pair of star-cross'd lovers take their life;
Whose misadventured piteous overthrows
Do with their death bury their parents' strife.
The fearful passage of their death-mark'd love,
And the continuance of their parents' rage,
Which, but their children's end, nought could remove,
Is now the two hours' traffic of our stage;
The which if you with patient ears attend,
What here shall miss, our toil shall strive to mend.
We know both Romeo and Juliet are from noble families from the very first line of the prologue since the households are described as being "alike in dignity." We also know that Romeo and Juliet are destine for fate since they are described as "star-cross'd lovers" - or lovers that are full of bad luck. We also know that Romeo and Juliet tragically die since they "take their life." Since Romeo and Juliet are "star-cross'd" and commit suicide, the audience will feel a sense of pity for them, especially considering their ages.
This then leaves us with Romeo's tragic flaw - he's imprudent. In other words, he acts rashly without considering the consequences of his actions. The play is full of examples of Romeo's rashness - from his short-lived love for Rosaline and his love for Juliet to killing Tybalt, Juliet's beloved cousin. In terms of his short-lived love for Rosaline, we see Romeo as a character that we might not be able to take so seriously since he moved from being madly in love with Rosaline to madly in love with Juliet within a couple of pages. When Romeo and Juliet find out that they are their families' worst enemies, Romeo continues to pursue the relationship anyway. In fact, they end up be marrying rather quickly. This is the point where Romeo decided and is responsible for his own fate - he didn't have to marry Juliet. Finally, the Prince warned that if there was another fight, that the instigator would be punished; thus, Romeo was banished from Verona, forever changing his nobility status.
With that being said, I think there is an argument to be made that Romeo is, in fact, a tragic hero.
However, to play devil's advocate, so to say, an argument could be made that the audience may not feel a sense of pity for Romeo since we realize his is imprudent from the very beginning with his "love" for Rosaline. In other words, it's hard to feel sorry for someone that makes the same mistakes over and over again.