In the poem there is not a single time where it is said that the person is a doctor except of some minor hints and the title of the poem. So how can you be sure that the speaker is indeed a doctor?
The title of the poem “A Doctor’s Journal Entry for August 6, 1945” itself is the single most important element to indicate that the speaker in the poem is a doctor. When the poet himself mentions it in the title, we have no need to find out proofs in support of the same.
The poet Vikram Seth actually narrates the events from the perspective of a doctor to make certain things look more realistic. Some subtle hints like dislodging a piece of glass by himself, giving confidence to his wife by saying “we’ll be fine” even when he himself is panic-stricken, quickly deciding that they must go to the hospital, or even maybe feeling no shame in a state of nakedness — all go well with the fact that the speaker is a doctor.