Coop games with Support/Healer Class

TF2 ?
(it has a healer, but I don't know if it is a good class)

Killing Floor/KF2.

Brink (quite desert now). Dirtybomb, Paladin
this is just MMO i play

Borderlands 2 Nurse Maya builds (although it is not that particularly effective there, sheer dps can get you a lot further). Plus it takes MMORPG levels of time to reach a decent OP8 build and get good enough gear for that level to raid with.
Most any of the Battlefield games, especially before it turned into a shallow spectacle shooter. Although it has become so much of a CoD-like series—where the sheep just migrate to the new game the moment it comes out, leaving behind desolate servers—that I have no idea if anyone even plays the old ones. Granted, Battlefield 1 (as in, the latest, 14th game) also has a dedicated medic role.
Unreal Tournament '99/2004 and even UT3 and UT4 can rely a lot on support team members in team-based modes (especially CTF), but since most people either play DM or just create one clusterfuck on the most basic ONS maps, finding an actual match and a team with enough functioning brain cells to actually play support is next to impossible.
Evolve has both a medic and general support class. It also has a ton of idiots and a community that seems to be constantly escaping the sinking ship.
Dead by Daylight has a survivor type who acts like aid, but not really for others, if it makes any sense…

I understand your predicament, but as much as it pains me to write this, but Overwatch seems to have the best choices right now for players who feel the most comfortable in the team support role. Blizzard successfully stole and combined the best ideas and it is the only game I know of where you can be a pure support.
You may try Paladins though. It is a very thinly veiled Overwatch rip-off/clone, but maybe different enough for you…

Try Insurgency

I always liked playing the medic in Battlefield.

Evolve, Alien Swarm

perhaps Vermintide?

edit: how could I forget Natural Selection 2

I'm always playing a healer in Space Hulk: Deathwing. And the game is now much better than it was at launch :)

I would say Dauntless?
It's currently in closed beta and you can only get access to it through the founder's pack however, it will become a free 2 play game once it's ready (if i'm not wrong).

Its something similar to monster hunter, though I have never played it myself it looks interesting and has a medic role in it. It is a 4-player coop game as well.

bump

Master X Master (MXM)

Unlike LoL, Dota and HOTS it has a PvE mode. You can do dungeons (up to 3 players) with many difficulty ranks and there's also another mode where you need to defend a base against the waves.

It's Free to play and there's a good amount of support heroes (heals, buffs, shields, crowd control... you name it).

Sounds like the good ole days of

1. Return to Castle Wolfenstein
2. Wolfenstein: Enemy Territory
3. Enemy Territory: QUAKE Wars

Please note that Co-op (ie, Team PvE) refers to a mode where multiple players play against an environmental or AI opponent (and is mostly what you'd encounter in an MMORPG, excepting ones primarily based in PvP). Team PvP (which covers most MOBAs, most online FPS games, etc) is the mode wherein a team of players competes against another team of players.
If you're looking for games of either mode, you can simply state "team games", as that is the overall category.

I only note the distinction because right now I'm uncertain if you're indicating a preference for a certain flavor of games (as you mention a wide variety, but you also seem to be indicating disinterest in the pvp ones), or if you're trying to indicate that you're open to any team game with a healer class.

Some Diablo clones could do the trick (Torchlight, Path of Exile, Titan Quest, etc.)

Warface in coop mode is exactly you describe.

Warframe? Is MMO-ish, has lot of grind, but the gameplay is very different. Has at least one "dedicate" healer class.

Battleborn, not sure on Path of exile, maybe some Borderlands builds.
Maybe Victor Vran, Grim Dawn, Diablo 3, Divinity Original Sin 1 or 2, etc.
Moon hunters?

Orion Prelude and Guardians of Orion both have the support/healer class and are online co-op. They don't have favorable reviews though.

I'm not sure about Sanctum 2 but who knows? I know only it have pve, coop and tower defence.
Try Killing Floor 2. I think it is best choice (I don't try this game).
I tried Dark souls by "healer" but I didn't find gameplay very interestring.
Just pve-coop as I know - Don't Starve Together or sandbox Space Engineers.
If u agree to try pvp-coop how about Dead by daylight cause it have too much coop-help to survive from player as serial killer?

For critical situation: yes-yes, I know mmorpg is bad option but I find pve-coop-healer gameplay in small cozy family guilds in... I'm sure nobody not guess xD ... in old-story which called World of Warcraft

If you like RPGs, you might be interested by Lord of the Rings : War in the North. It's coop up to 3 players.
You have Farin, the Dwarf, a tank/berserker/crossbowman; Eradan, the human, a hunter/rogue/swordman, and you have Andriel, the Elf chick, a caster/healer/support.
But clearly, she's not a white mage, you won't cast dozens of healing spells. She casts a sanctuary, a dome which heals the allies inside and deflects the ranged attacks. I think she has also a pure healing spell, but I might be wrong.

I enjoyed this game, it's quite entertaining with friends.

I'm gonna throw my hat in with Paladins. I love it, though it's still in 'beta' with frequent changes, some of them significant, and frequent bugs to go with.

Monster Hunter World is fun to play as support with Hunting Horn. You have to fight the monster to deal damage and your abilities are usefull. The basic is to boost capacities (attack, hp, resistances, etc...) and you can cure (heal, remove poison, etc..) with the appropriate equipment. You can easily play with friends or random people.

And perhaps Divinity Original Sin 2, I never played but I see there are healing spells and coop.