1. He does a good deal of public work, of which he did not specify the nature.
2. He does a good deal of public work, the nature of which he did not specify.
3. He does a good deal of public work,whose nature he did not specify.
4. He does a good deal of public work of which he did not specify the nature.

Question: Which is/are grammatically correct?
Do they have the same meaning?

Question

1. He does a good deal of public work, of which he did not specify the nature.
2.  He does a good deal of public work, the nature of which he did not specify.
3.  He does a good deal of public work,whose nature he did not specify.
4. He does a good deal of public work of which he did not specify the nature.

Question:  Which is/are grammatically correct?
Do they have the same meaning?

Answer

All are grammatical.  With a comma, I interpret the subordinate clause as referring back to the entire noun clause "a good deal of public work".  Without a comma, I interpret it as referring back only to "work". Others might not make that distinction. The sentences do not state whether he does not specify the nature of ALL of his public work, or just SOME of his public work. That need not be a problem.
In #1 "about" works better than "of".
In #4 (same as #1 except for comma) "for" works better than "of".

Answer

Sentences 1 and 4 are grammatically incorrect. Sentences 2 and 3 are grammatically correct. In sentences 2 and 3, the relative clauses reflect back correctly to "public work."