I'm a fan of D20 systems, but the way hit points works always seemed kind of odd. The characters obviously can't develop an ability to take more stab wounds as they gain experience. So that means that all these "hits" are really misses. So what's the difference between a "hit" miss and a "miss" miss? Is there a system that handles this more realistically?
There's a Few
Shadowrun, GURPS, and New World of Darkness are all noted for their lethal and 'realistic' combat systems to various degrees. Certainly all three games treat combat as being a serious affair in which people die and injury is to be avoided at all costs.
But all systems abstract to some extent
There's no such thing as an absolutely realistic system of combat short of grabbing actual weapons and hurting each other (I do not suggest this: it's bad for your health). All RPGs abstract combat, injury, disease, and poison to an extent - how much is the only real question up for debate. If your group is happy with the current level of abstraction there's not a lot of reason to change it.
Also, that's not the only interpretation of Hit Points
Hit points are a lot of things to a lot of different people in D20 systems; consider another form of abstraction that might work better for you. Or buy into the "characters become superhuman" idea, which can be an awesome trope to explore.
d20 hit points do not have to represent actual damage
Your hit points measure how hard you are to kill. [...]
What Hit Points Represent: Hit points mean two things in the game world: the ability to take physical punishment and keep going, and the ability to turn a serious blow into a less serious one. (D&D 3.5 PHB1, page 145, and the d20 SRD)
What that might look like is left as an exercise for the individual group/player.
The d20 System engine also provides wounds and vitality as a modification of the system, which you may find to be more to your liking. It's designed to be more cinematic, although whether it succeeds is arguable. What it does accomplish is to provide separate linked minisystems for "being worn down" and "being actually hurt," with one acting as a buffer for the other which can be bypassed under certain circumstances.
Other systems have a wide range of combat-consequence mechanics
Other answerers have mentioned several systems already. The vista of RPG engines is vast, and many--perhaps most, by now--deviate wildly from the traditional "hit point" system. Some are unashamedly narrative, while others are brutal called-shots systems.
Which of these is "most realistic" is up to individual taste.
As has been pointed out elsewhere, all RPGs are an abstraction at some level. We couldn't model the universe perfectly in anything less than real time even if we understood it well enough to make that possible. Choosing an RPG engine, then, is tantamount to a statement of intent: it declares what is valuable to us in the game experience.
Our impressions of what combat is "really" like might be vastly different depending on our personal experience and exposure to violence in the media. There's an RPG system for every iteration of that paradigm, if you care to dig, but which is truly "most realistic" is a lot less objective than it initially seems.
You'll have to consider what it is about combat damage that you want to be realistic, what you're willing to sacrifice to achieve that experience. Finding the system which models that well without sacrificing some other part of the RPG experience you also value will be the challenge: the more complex the system, the longer turns will take and the less emphasis will be put on other elements of play.
Rolemaster used to be quite realistic. Ok, you still won many HP with experience, but the critical system made it possible to die on a single blow. So, hits that were not critical were supposed to be lesser wounds or small hits that caused fatigue, but no serious injury.
Chivalry and Sorcery (which people must be tired of hearing me recommend) has a combat system interesting in this regard. A hit does damage (reduced by armour) that first reduces your Fatigue points; you might consider this being winded. After the second or third blow, your fatigue is used up, and you start taking Body damage, which takes much longer to heal. After three or four more hits, you die. High-level characters will have better conditioning and endurance training, but your Body and Fatigue can never go above 125% of what you started with.
(If you think this is exactly what you are looking for, remember 'be careful what you wish for; you might get it'. The C&S combat system went through at least four revamps, with probably another one in 5th edition, due Real Soon Now. And even the simplest makes any sort of fight complex and time-consuming; the full version, with Advanced Initiative, Bash rules including Body Bash and Shield Bash, and graduated criticals, is best described as 'insanely difficult'.)