Can you run a gritty combat game with the 4e D&D rules?

In the discussion about my last post, there seems to be a split along the lines of people who favor grittiness in their combat and others about whether or not the sheer amount healing in 4e can reduce the tension of combat. Personally, I think the amount of healing makes the game lack suspense for the players…it’s not a matter of whether or not they’ll finish a combat, it’s just how long it takes. Sometimes I think the players are encouraged to use Daily abilities just to finish a long combat faster, because the auto-healing means that if they just sit and beat monsters down they’ll be no worse the wear at the end of the combat no matter how many blows they take.
Others are saying that 4e has just redefined the lines of where you begin to feel tension. You feel the loss of healing surges more than the loss of hit points, and there might be one combat a day where you wish you had a healing surge left. Some are just saying flat out they don’t think tension in combat does anything for the game.
So, what I want to know is, can you run a gritty, more realistic combat simulation with the 4e rules, or is there too much healing too easily accessed for it to ever feel like that?
I personally think it’s something missing in 4e…in previous editions, you had more flexibility to tweak combat to taste, but now since everything is so tightly wound into the idea of healing surges, even magic items, that it would require a major overhaul to make a grittier game possible
There’s only one individual combat per day where anyone should feel worried.