Let's fix the sentence, then fix the "rule."
Original:
A well-known medical specialist stated that it (is/was) physicians' moral duty to inform patients about the possible side effects of any drugs they (will/would) recommend.
Corrected version:
A well-known medical specialist HAS STATED that it IS A physician'S moral duty to inform patients about the possible side effects of any drugs (THAT) THE PHYSICIAN recommendS.
Analysis:
1 - / A well-known medical specialist HAS STATED /
2 - / that it IS A physician'S moral duty /
3 - / to inform patients about the possible side effects of any drugs /
4 - / (THAT) THE PHYSICIAN recommendS. /
1 - /A well-known medical specialist HAS STATED /
Use "has stated," if the action occurred during an unfinished time period.
Example: "... has stated today," "... has stated this week," "has stated recently," or "has stated" (recently assumed).
Use "stated," if the action occured during a finished time period.
Examples: "... stated yesterday," "... stated last week," "... stated on March 1, 2002."
More information about the simple past and present perfect here:
https://www.ef.com/ca/english-resources/english-grammar/present-perfect-vs-simple-past/
2 - / that it IS A physician'S moral duty /
Use "a physician's moral duty" to indicate a moral duty for all physicians.
Use the simple present "is" because the moral obligation is a general truth.
More information about the simple present:
https://www.ef.com/ca/english-resources/english-grammar/simple-present-tense/
3 - no analysis
4 - / (THAT) THE PHYSICIAN recommendS. /
Replace "they" by "the physician" for clarity.
Use the simple present "recommends" because recommending medicine is a repeated action.
More information about the simple present:
https://www.ef.com/ca/english-resources/english-grammar/simple-present-tense/
As for the "rule," that's a big discussion. I'll add a brief comment.