Melee should have lowest damage because they can take damage like a champ, because they go toe-to-toe and can survive it.
stealth should have bonus damage because they attack from the shadows and hit them in their weak spots not because their knives do less damage, in fact weapons with less damage should ,gasp, do less damage unless used properly like a sneak attack into a soft spot.
magic should be able to dish out a hell of a punishment if thats what you're going for or less damage but more effects like paralyzing and what not.
On the contrary, nothing YOU said makes any sense.
Melee should do the most damage because, guess what, swinging an axe into someones face is going to hurt them more than shooting them with a tiny sharpened stick, or even covering them with gasoline and setting them on fire (the equivalent of a fire ball spell, in terms of damage to skin and flesh) You can easily survive those last two, if you only get hit once. Not so much with the axe in the face.
Stealth should get bonus damage for close up hits, because, well, it's like any other form of melee, ie, more dangerous than arrows or spells. You compensate for the smaller size of the knife compared to the axe (which decreases range and momentum) with increased accuracy. It's easy to get a kill shot on someone with a knife if they don't know you're there, because you can target the throat, or heart, or other vitals. But if you come at them from the front, they can easily grab your wrist and disarm you...something harder to do when it's an axe coming down with all it's enertia.
Magic has no real world equivalent, so we have to compare to real world damage from similar effects (fire, extreme cold, etc) Covering someone in gasoline and lighting them on fire is fairly easy to do, just like a low level fireball spell. And just like a lower level fireball spell, it does less damage than an axe blow to the face. You might die eventually, but you'll live long after the fires gone out. Now, a flame thrower or a small bomb might kill someone relatively quickly, but that takes a lot more money and experience to buy/build/use, just like a higher level magic spell.
Magic should NOT be over powered, "just because it's magic." Magic should be BALANCED, because this is a game, and you need to make all things proportionate. Obviously magic is overpowered on it's own, otherwise mages would traditionally be given the same statline as warriors. But game makers for decades have realized "hey, these guys can do crazy $#*!, let's make 'em squishy.