GURPS does something like that. Damage isn't static in the way that the OP mentioned, but damage is based on character strength. Weapons then have a 'static' modifier to what the character can do.
For example, a small dagger might simply change damage from crushing (what unarmed attacks typically do) to cutting (for a slash) or impaling (for a thrust.) On the other hand, a big weapon like a battle axe might say something like "swing+4" in the damage entry. (Note that I'm not looking at books; this is simply an arbitray example to show how it works.) What that would mean in game terms is that the weapon would (with a swing) do whatever the character's swing damage is plus four more.
If you're looking to go completely static, that's not hard to do. Figure out what the average values of dice are and convert damage to that. For example, the average value of a d6 is 3.5; as such, a weapon which would normally do a d6 would do either 3 or 4 damage (depending on if you round up or not; most rpgs assume rounding down except in specific cases.) For critical hits, you could either multiply the base value in the manner of D&D 3rd Edition; so a d6 weapon with a x2 multiplier would do 7 damage on a critical hit. Another option would be saying that normal hits do average values, and critical hits do full die values. In this case, the d6 weapon would do 3 damage on a normal hit and 6 damage on a critical hit. To give other examples, a d12 weapon would do 6 on an average hit and 12 on a critical; a 2d6 weapon would do 7 on an average hit (3.5 + 3.5) and 12 on a critical. Depending on what you're looking to do, that might be desirable or not.
Yet another option would be to just apply the multipliers (if you're using them) to the average value. A x2 weapon would do a full die of damage, so 3 for the normal hit of a d6 weapon (like mentioned above) and 6 for a critical; 9 damage for a x3 weapon that uses a d6, and 12 damage for a x4 weapon which uses a d6.
If you're looking to go even more simple than that, max the damage value on a critical and add whatever the modifier is. In this case, a d6 weapon with a x2 multiplier would do 3 damage on a normal hit and 8 (d6+2) for a critical; a x3 multiplier would be 3 damage for a normal hit and 9 (d6+3) for a critical.
...and I just realized while typing that both methods lead to pretty much the same thing, so using the second one is probably more user friendly.