One of the statements below contains a word used incorrectly. Choose the option which has the incorrect or inappropriate usage of the word.

In the given sentences, the word " state" has been used. "State" can have multiple meanings based on the context. It can refer to a nation or territory, or it can also refer to the particular condition someone/something is in at a particular time. It can also mean to express something clearly. 

In case of Option A:[The judicial magistrate asked the prosecuting lawyer to state the grounds on which the defendant had been arrested.] "State the grounds" means to express or specify the reasons. Here, the usage is correct. 

In case of Option B:[The state of his grandfather's health was critical, but it was difficult for him to get in touch with all the members of the family.] "The state of his grandfather's health" refers to the current condition of the grandfather's health. Here also, the usage is correct. 

For Option C: [Uttar Pradesh, being the most densely populated state in India, has the highest number of representatives in Parliament.], Uttar Pradesh is a state in India; the word here refers to a political/geographical region. 

Option D :[Her admirers considered her demeanour to be of great state and refinement, and most impressive] however uses the word "state" awwardly. "state " is used here as a noun, which typically does not imply "dignity" which is the intended meaning here.

Hence, option D is the correct answer in this case.