What is the Islamic ruling if a woman wants to go out and study for her own growth and development (which in turn will no doubt make her a good mother and wife) but the husband doesn't want her to due to the Hadith that says women should stay in their homes and not come out unless very necessary.

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

There is no doubt that the best job that befits a woman is to stay at her house in order to manage its affairs and to take care of her husband and children.
But, as you stated in your question, if studying will develop her character as a mother and wife, and if the Muslim society is in need of such a study (medicine …), then there is no blame on her. Moreover, here are some conditions that are to be met in such a state:
1) A woman should have her husband's consent.
2) She has to wear the complete Islamic Hijab.
3) She has to avoid mixing.
4) She should give priority to her house and family.
Allah knows best.