Had any 20th century navies used damage-control specialized vessels in battle fleets?

As far as I know no damage control ship comes even close to matching the speed of a carrier battle group. That would mean tugboats that can do over 30 kts. No tugboat in the world can do that. They have the horsepower alright, but they use that power to pull ships. Not for speed.

It wouldn't make any sense building a tugboat that can do +30kts to keep up with a battle fleet. That ship would be hugely expensive. A far easier and much cheaper solution is to construct naval ships that can assist other ships. For example, that's how destroyers are designed.

For example after the Forrestal fire a destroyer approached the carrier and used its onboard fire hoses to quench fires ...

That's correct, many naval ships have enough resources build in to be able to help other ships. The bigger ones can even tow large ships. They have been build to assist other ships, but they haven't been designed as specialized damage control ships.

The navy can do most of the assist work in the battle zone, and tow a damaged cruiser or carrier out of harms way with naval ships. Once the damaged ship is in safer waters, a tugboat can take over.

Every combat vessel has an assigned damage control role as well as internally a damage control division of specialised crew (at least on vessels large enough to warrant one), and every crew member has at least basic damage control training and usually first aid training as well.

This has been the case for a very long time, though it often wasn't drilled very frequently and the abilities of most crew members tended to be rudimentary. After the Forrestal and Enterprise fires the US Navy started to take this a lot more seriously and greatly increased its emphasis on such training, raising standards significantly.

As to tugs, pretty much any ship can act as a tug in an emergency, and destroyers and cruisers are frequently used as such in case of accidents.

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, and counting on the assisting vessel being nearby, and the 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, because bringing your ship so close to a stricken vessel is by no means something to be taken for granted.

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.

I'm fairly sure this has never happened. Ships capable of carrying specialised repair crew and equipment certainly exist, under such names as tenders, repair ships, depot ships and mobile bases, but they lack the speed to accompany a battle fleet. Creating such a fast repair ship, and giving it adequate defensive armament to prevent it being a complete liability in battle would be extremely expensive, and its usefulness would be limited.

It can only help one or two ships at a time, and many ships may be damaged in a battle. Also, naval vessels don't go around in groups of constant composition. Their groups are frequently split up, joined in different configurations, and generally used in unexpected ways. It thus makes more sense to give all of them some intrinsic damage-control capability, and allow ad-hoc co-operation when it is needed.

Addendum: the deliberate design of combat ships with very limited damage-control capability is a recent innovation, in the Littoral Combat Ship family, and many commentators think it's a serious error.