Melee healers are definitely possible in the technical sense, and I made a brief outline when people were speculating DNC might be a healer as well as when some folks were calling for a darker, edgier healer. I recall seeing some other people lay out some frameworks or themes as well.

Personally, I think melee healers would be a good candidate for tying healing to damage in some way. Like a long-cool down that let you convert a portion of damage into healing or a buff to healing, perhaps draining the enemy targets in some way. The job would need to keep moving like a ninja, monk, dragoon, etc. which could involve a mini-game of when "cast while moving" vs. "stationary cast" options are available to make sure party member didn't die while their healer was dodging mechanics. Another solution would be strong shields or regen to cover that movement time, but that's already readily available in other healers.

Some people have chimed in for Chemist as healer, which could be modified to either primarily ranged or primarily melee in its reimaging, or even a hybrid. Lots of other jobs from other FF games or new types for IV could be on the table as inspiration for a new healer as well. But, SE has by their own admission been struggling for a couple of expansions on the healer role issue, so I don't seem them in a rush to push for a new healer job even though the next window would likely be 6.0.

Though if they want to balance fighting jobs by number, that would mean another Physical Ranged DPS and another Healer in the next expansion to make an even four of each category