Yes, you can play the whole game without a healer. Easily. I never bothered having a healer in my party in that game, because I found Wynne a bit boring as a character. So I'd usually go with me (a rogue), Sten, Shale, and usually the dog. If you make Sten a Champion he gets an AOE stun shout. So does the dog. So does Shale. So fights usually broke down like this: CHARGE SHOUT (everything's stunned) DAMAGE DAMAGE DAMAGE rinse repeat. On the rare occasions when my party members ever got hit, I'd just down a health poultice and move on.
It's even possible to do this in DA2.
Granted, I don't bother playing nightmare, but I know people on the Bioware forums who solo'd nightmare mode in Origins on a rogue and never used a single health poultice. So it is certainly doable.