The Rukhmabai case has been written about by historians of British India on several occasions. Antoinette Burton published an essay on it, for instance, in 1998: here. Sudhir Chandra also had something about it in EPW in 1996: here, and there is a full chapter of his book, Enslaved Daughters, online on the open internet here.
In brief summary, Rukhmabai was married in a child marriage. Her family facilitated delaying the consummation of the marriage for several years, during which time she was educated. She started posting letters in the Pall Mall Gazette, and then she sued to have the marriage dissolved. The judge initially sided with her, but then reversed the decision.
Civil & Military Gazette
April 8 1886
The Chief Justice of Bombay and Mr. Justice Bayley have delivered judgment in the now famous case of Dadaji-Bhikaji v. Rukhmabai for the restitution of conjugal rights. The details of the case have more than once been fully explained here. On the first hearing in September, Mr. Justice Pinhey dismissed the suit with costs, in words which reflects the greatest credit on his sense of equity, albeit he was wrong in point of law: 'It seems to me that it would be a barbarous, a cruel and revolting thing to do, to compel a young lady under these circumstances, to go to a man whom she dislikes.' All India by this time knows the circumstances under which the unfortunate girl had been placed. A husband was chosen for her whom she felt no love on reaching years of discretion, and with whom she declined to live as a wife; preferring rather a life-long widowhood. Some months ago there appeared in a Bombay paper letters written by a "Hindu Lady" -- touching on the horrors of infant marriage. These letters were from the hand of Miss Rukhmabai. She was educated above the level of her fellows, and strove earnestly for the knowledge which is the natural right of a woman in the West. 'Without the least fault of mine,' she complains in one of the letters above mentioned, "I am doomed to seclusion. Every aspiration of mine to rise above my ignorant sisters is looked upon with suspicion and interpreted in the most uncharitable manner." Before Mr. Justice Pinhey, Miss Rukhmabai alleged, as her reasons for refusing to live with the person who had chosen for her "(1) his entire inability to provide for the proper maintenance and residence of himself and his wife, (2) the state of the plaintiff's health in consequence of asthma and other symptoms of consumption, (3) the character of the person under whose protection the plaintiff was and is living in the house in which he called on her to join him." [...]
If Miss Rukhmabai refuses to abide by that decision, she can be imprisoned for contempt of court for six months. If her appeal succeeds, and the legislature decides that marriages contracted for children cannot be enforced, if the parties object to them when men and women, we may safely hold that infant-marriage in this country is doomed. the self-interest of the parents will prompt them not to throw away dowries on infants who may rise up later on, and call their matchmakers anything but blessed. the whole case furnishes a convincing instance of the utter rottenness of the law; and is sufficient answer for those who clamourthat the East and West shall be treated on an equal footing. A society that tolerates such a law has been expounded by the Chief Justice of Bombay to the unfortunate girl who sought relief, is a society that places itself, by the cowardly cruelty of that very law, below all civilisations. Not all the decrees that were ever engrossed on parchment, or all the suave speeches ever delivered to 'a might nation kindled with the fervour of the West and struggling to be free,' will disguise the fact, that the bases of the society which is to be painfully hoisted to the level of Western ways and thoughts, are laid on ignorance, corruption and oppression. If there be--as the native periodicals assure us daily there are--men who regard with disvaour any attempt at political reform on the part of the Hindus of India, our contemporaries may be certain that the case we have just quoted will prove a powerful weapon in their hands.