Not to my knowledge. John Dallman has already pointed out reasons for this, I wanted to add another one.
One of the issues you would face with dedicated "damage control vessels" is this:
Any ship so heavily damaged that its own crew and equipment are insufficient to control the damage and in need of help is, very likely, also very unsafe to approach by another vessel. It might be no longer under controlled steering, it might be on fire (with accompanying danger of explosion), it might be listing heavily and / or in danger of sinking.
Approaching another vessel on the high seas in a way that allows transfer of crew is not a trivial task between two fully functional vessels. Planning to do so with one vessel by design being in a very dire situation, the assisting vessel being nearby, and counting on that approach actually working, would be wildly optimistic.
You quoted the fire on USS Forrestal, and how USS Rupertus approached the stricken carrier to assist with the firefighting. You should also note that this action was called "an act of magnificent seamanship" by Rear Admiral Lanham...
If a nearby ship is in a position to render assistance, it will (attempt to) do so -- and the ships already in a battle group, especially the smaller ones (destroyers and cruisers), are already sufficiently equipped.