Why is there "a Romeo" but not "a Juliet"?

A Penelope is defined as "as faithful wife" (here). It can probably be used more loosely to mean any faithful female lover. It comes from ancient Greek mythology; Penelope was the wife of Odysseus.

Unlike Romeo, which often (but not always) suggests that the male pursues multiple women (cf. Don Juan or Lothario), Penelope suggests a woman who is faithful to a single man.

As to the question of why "Juliet" didn't take off like "Romeo," it's hard to say. I doubt that any answer would amount to more than armchair speculation.

It is the atypical nature of a character that makes for an informative literary eponym.

In this context, I think the thing that makes Romeo remarkable is not the fact that he fell in love with Juliet but that he is first introduced as being madly, passionately in love with someone else. It is only while trying to court Rosaline that he meets Juliet. He again falls madly, passionately in love, promptly forgetting about his previous infatuation. Indeed, based upon his mood and the apparent ignorance of his cousin, it seems he is only recently in love with Rosaline herself.

It is my understanding that it is this tendency to fall easily in and out of love that is being commented upon when someone is described as "a Romeo".

In a broader context, Romeo and Juliet are most remarkable for the doomed nature of their love, but this has given rise to terms describing the couple, rather than either individual, e.g. "star-crossed lovers". Another interpretation would be that Romeo and Juliet are remarkable for the depth of their love. However such an interpretation would seem to contrast with and belittle one's own experience of love. Rather than diminish their own relationships, I suspect that most people imagine that they love their partners as much as Romeo and Juliet loved each other, and that the tragedy of Romeo and Juliet is driven more by intolerance and cruel fate than by some inherently atypical nature of their love.

In this respect, Juliet's love for Romeo would not be remarkable enough to coin a new term.

I question the premise of your question. Perhaps you are just looking in a skewed dictionary:

Juliet

The perfect girl. She will light up your life from the moment you meet her.

She's smart but not nerdy, hot but not slutty. Beautiful body and a gorgeous smile, and always up for a good time. A Juliet will be the best girlfriend/friend you will ever have, she's the girl you will want to make your wife.

Sexy, athletic, intelligent, loving, and knows how to party. She may seem intimidating, but that is only because she knows what she wants, and knows she needs a real man.

Sure, the source above is Urban Dictionary, but at the very least it reflects actual usage to some extent. The entries are created by users, and there are several entries for Juliet, which might indicate a significant amount of common usage.

As I mentioned in my comment, I've been searching ngrams and the Internet in general in different ways, and despite the situation in the quoted dictionary, there seem to be plenty of uses of "my Juliet" or "she's my Juliet", etc., and it doesn't seem like using Romeo in the same way is significantly more common.

Also, Juliet (from Wiktionary)

3. A woman who is or is with a great lover.
4. By analogy with the Shakespearean character, a woman who is in love with a man from a family, party, or country opposing that of her own.

It seems like the "reputable" dictionaries that come from the age of print do not have a definition for Juliet equivalent to their definitions for Romeo, but the crowdsourced online dictionaries do. Perhaps the culture on the two terms is currently in flux, and the older dictionaries are not caught up.

Interesting source (Bartleby): Juliet

Daughter of Lady Capulet, and “sweet sweeting” of Romeo, in Shakespeare’s tragedy of Romeo and Juliet. She has become a household word for a lady-love.

Asking "why isn't there X" is often unanswerable. Why isn't there a word for female cousin distinct from male cousin, like brother is distinct from sister? Why isn't there a good gender-neutral pronoun we can use for humans? etc.

In this case, I'd hazard that the answer is "sexism", specifically, social attitudes that praise a man for chasing women but scold a woman for chasing men. A seducer of women is a Romeo. A seducer of men is a slut. All women are expected to be gentle, devoted lovers to their husbands. Women were expected to be virgins on their wedding day. Men, not so much. Maybe if enough English-speaking people learn to stop slut-shaming, we'll eventually agree on a positive word for a "female Romeo".

To make even a plausible guess at the answer to this question, you would need to learn an enormous amount about the sociology and history of the English-speaking world, from the 1590s through today. Here are just a few examples of things that might be relevant.

• In many societies, past and present, women and men were and are expected to approach romance in very different ways. The evolution of language reflects this.
• Was English Renaissance theater largely a guy thing? We apparently know very little about Elizabethan theater audiences, so I guess it could have been. There aren't even any contemporary paintings you can turn to for a glimpse of audience demographics: a painting the V&A calls "one of the first attempts to depict the interior of the first Globe theatre" was done 150 years after the fact, and its painter basically just made everything up.
• Although Juliet hasn't become part of English in the way you suggest, she has left her mark in names like Juliet cap and Juliet balcony. These names, however, were coined much more recently than Romeo. The OED shows Romeo being used in the way you describe as early as the mid-1700s, and possibly even as far back as the mid-1500s—before Romeo and Juliet was written! (Wait, how is that possible?) On the other hand, the OED only shows uses of Juliet cap starting in the early 1900s, and it doesn't even have an entry for Juliet balcony.