I'm honestly not sure the game could actually manage not bringing out a new healer. New jobs are what sell the expansions because they're not only the easiest thing to explain to people about an expansion, but also the one part of an expansion that is fully tangible to players as the lenses through which we see the game. Those lenses for healers will be pushing eight years old by the time 6.0 rolls around, and without a new job to revitalize the hardest role in the game (and maybe some other incentives offered), I'd genuinely expect FFXIV healers to wind up on the endangered species list.

As for next healer, no it won't be a pet job because scholar takes that spot (more or less), it won't be melee unless it's only for doing damage, and it'll most likely follow the same template that every other healer job works off of. Something like a Medic or any other alchemical-themed job is a possibility, but not with a pet.