You have a valid reason to request to remain at your current place of posting based on humanitarian and medical grounds. Your wife is currently under the medical care of AIIMS doctors, and this specific medical facility is not available at the proposed place of transfer.
If the promoted position is available at your current place of posting, the authorities may consider extending your posting for a specified period. While transfers are generally based on administrative exigency, in special and unique situations such as yours, this exigency may be relaxed.
To pursue this matter, it is advisable to submit a representation to the competent authority, urging them to consider your request. If your representation is rejected without a valid reason, you have the option to approach the High Court under Article 226 of the constitution for the issuance of a mandamus writ. In such a case, the High Court may direct the authority to reconsider the matter afresh in accordance with its directives.