EU/EC261 Due?
 

What are the specifics? Without them, it is impossible to answer your question at least answer it correctly. In addition to the routing and carrier specifics, what was the reason for the delay?

AA first legt, EY rest.
They stated "medical reason"
US-Paris-Auh-Bom
Flight delayed from Paris. As a result missed connection in Auh.

If said Medical Reason was a passenger or crew on the plane situation I believe you're SOL. It can be tricky if it's true ill before boarding and unable to find sufficient crew which if it only was delayed at a limited service outstation it is less likely... but no harm is asking.
Just send off an email stating the flight info and requesting compensation for delayed flight according to the arrival delayed time. State that medical exception for a delay must be completely out of the airline's hands and unless said delay is because of something within scheduling control of the crew you expect compensation.
Keep it short, simple but with more info than you've given here to get advice.

1. I suspect that an actual medical reason on EY at CDG, even if it is a crew member, will be an extraordinary circumstance and mean no compensation.

2. More important, but EY is not likely to figure it out is that EC 261/2004 does not actually apply to this ticket. For the very reasons stated in the decisions which have led to the determination that the Regulation is determined at the final ticketed destination. This is a single ticket US-BOM which happens to have connections at CDG and AUH. As various courts have pointed out, those connections are mere mechanical and administrative matters, not relevant to the passenger's actual ticket. Thus, this ticket is on a non-EU carrier between two points outside the EU and the Regulation never applied in the first place. Now, that would not deter me from making the claim (recognizing that #1 will likely kill it). Someone at EY would have to go back and analyze how OP got to CDG,