Why does 他没看书 use 没 and not 不 in this dialogue?
The opposite of &quot;他没看书&quot; (he did not/has not read books) is &quot;他有看书&quot; (he did/ has read books)- 没(has not) here replaces the adverb 有(has)- to indicate 'absent' of the action
The opposite of &quot;他不看书&quot; (he does not read) is &quot;他看书&quot; (he does read)- 不 here modifies the verb 看书 (read) into its negative form 不看书 (does not read) - to indicate 'refusal' of the action
Edit:

A: 喂你在做什么呢
B: 我在看书呢
A: 大卫也在看书吗
B: 他没看书。他在学做中国菜呢

Since A asked B &quot;what are you doing&quot; and B replied &quot; I am reading&quot; and then A asked &quot;is David also reading book (implying 'now')?&quot; we can conclude B's answer of what David is doing should be in the same tense as the answer on what B himself is doing, and that was 'present progressive'

The correct answer should be &quot;他没(在)看书。他在学做中国菜呢&quot;*

&quot;他没看书&quot; doesn't indicate specific tense (it can be past or present because 他没看书 only states the absent of the action), while 他没在看书 also only indicates a progressive/ continuous tense.
One thing I stated many times, Chinese verbs do not have tense, 没看 and 不看 do not indicate any tense, while 没有看 indicate 'perfect tense'; 没在看 indicates 'progressive tense'
The reason we know it is in present progressive tense, is we learned it in the context that A is asking B in present tense.
If B answered &quot;他不看书&quot;(he doesn't read - refusal of the action) then it would mean David never read book, and the sentence would be in present tense.