Das Mädchen ist zu einer Frau geworden.

English Translation

That girl has become a woman.

Is “zu” necessary? “Das Mädchen ist eine Frau geworden” Would that be a correct sentence too?

Yes.

No. But it can subtly add nuance and slightly change the meaning. See this link:

If you use “zu”, you lose the current state to gain the new state.

Example: The girl is now a woman. She is no longer a girl.

If you don’t use “zu” as in your alternative, you (additionally) gain the new state without saying whether you lose the current state or not.

Example: Die Frau ist eine Lehrerin geworden. She is now a woman and a teacher. She didn’t stop being a woman once she became a teacher.

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I think this could be explained in short:
X werden - to become X
zu X werden - to turn into X

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