Yes, a D&D party can function just fine without a dedicated healer.
I've played D&D in various forms for the better part of two decades, from 2nd edition, to third, to a high variety of d20-sytem / OGL variants, and not once has the absence of a healer rendered a party unable to adventure. And this was in systems prior to 4th, which sped up the "natural" healing available to non magic-users from a paltry 2 hp per day to a considerably quicker pace.
In 5th edition, where all characters can spend hit dice over a short rest to gain HP and characters get all HP back after a long rest, not having a healer will, at worst, formalize the same "camp after a tough battle" strategy which most D&D campaigns employ.
You may need to adjust a published adventure which was written with the assumption of a dedicated healer, but this would be true as well if you lacked an armored fighter, arcane spellcaster, or someone proficient in lockpicks.
