World of Warcraft Mists of Pandaria: Mistweaver Monks, if they take certain talents for a build called "Fistweaver" by the community, they will spend most of their playtime punching enemies while passively healing the party, with minimal standing around casting sparkle heals on people. I don't think there was an explanation for how exactly punching enemies also heals your friends.
Final Fantasy XI: there are two ways to be a martialist healer. The first way is to main a martialist class like samurai or dark knight, and to then equip the dancer job as a subjob. Dancer gives you access to cheap, 100 TP instant abilities (TP being the resource for martialists, as opposed to MP for casters) that can heal a target or the party and cleanse debuffs. One of the Dancer abilities applied a debuff to the target that caused any party member who attacked that target to get healed. The player does not have to stop to cast a spell, nor is he prevented from using his 1,000 TP offensive weaponskills. However, your heals will not be as powerful as a maining a healer class, so your party will either need to pull weaker mobs, or be constructed to be tankier and/or have other players who also bring healing abilities (or bring healing potions, often overlooked). The other way is to main the Puppetmaster job. The player becomes a fistfighter with a little robot pet at their side that can be customized to either maximize damage, tank, or cast healing spells.
A Puppetmaster and accompanying automaton.
Trails into Reverie and Trails Through Daybreak 2 each have one swordsman party member with a swordmanship ability that stabs the enemy and also heals the whole party (as well as giving everyone CP, allowing them use more expensive weaponskills). This is quite effective, and when you jack up that character's speed and give them CP regenning equipment, they can spam this move every turn very fast and put out powerful healing, only needing to stop attacking to revive someone if that happens (though your party will occasionally want to cast spells to increase evasion). I was able to beat both games on the highest difficulty with a full party of martialists this way. However, there is no lore explanation behind these techniques, so it can feel a little arbitrary that this one mid-tier swordsman can swing his sword at the enemy and that will somehow, inexplicably, heal the party when all of these other swordsmen with much higher training can't.
Not pictured are other characters getting healed from this magical stab.
Guild Wars End of Dragons: this expansion added the Untamed elite specialization for the Ranger class. During the launch patch, Untamed rangers who wielded a two handed hammer and using a certain combination of equipped abilities, traits, and stats could become extremely powerful healers while whacking the enemy. IIRC the way this worked is that the player would summon ranger spirits that would passively heal the party, and then during the Untamed's rotation he would drop a puddle of water on the floor, and one of the hammer abilities would swing down and "blast" that puddle of water, healing the party. So I suppose there is some magic involved. It's not explained. Rangers can also get pets with the ability to heal the party, though the healing was insignificant so you could pick whichever else you wanted. This was strong enough to be viable in the 10 man challenge mode boss fights. Unfortunately a year later the ranger spirits were reworked, and then Untamed got a rework, so I don't think this build is possible anymore.
Warhammer Age of Reckoning had the Warrior Priest class who had to whack enemies in melee to build up charges cast healing spells.
Genshin Impact: The swings of the swordswomen Jean and Noelle will also passively heal you (for Jean it is a trait you have to unlock that gives her attacks a 50% chance to heal, for Noelle you have to press a button to use an ability that causes her attacks to heal). Again, no explanation for how this is supposed to heal people.
Lord of the Rings Online: Beornings while in bear form have the Relentless Maul ability that also heals the party, though how exactly this is supposed to heal the party isn't explained. LotRO calls its HP bars "morale", so perhaps swordsmen are supposed to be filled with courage by seeing their allied bear clawing orcs? Why aren't these swordsmen being filled with courage seeing each other fight orcs? For reference, the baseline healing class Minstrel would play songs to raise people's morale/heal.
Somehow, this will restore your HP bar.
