No, melee is not a damage type, and there is no straightforward way to resist melee attacks. D&D 4e doesn't divide up its damage types like that.
Instead, D&D 4e has damage types like Fire, Poison, Psychic, Sonic, Necrotic, and so on. Various effects and powers which deal damage (be they melee powers or something else) may specify a damage type. If they do, you take damage of that type. If they don't, the damage has no type at all.
There is an exhaustive list of damage types on page 55 of the Player's Handbook, or page 114 of the Rules Compendium. In the Player's Handbook, they're listed as: acid, cold, fire, force, lightning, necrotic, poison, psychic, radiant and thunder.
Your Diamond Bracers can resist any specific damage type, but you cannot choose to resist damage of no type - because that's not a damage type, it's the absence of one!
So how do I know what damage type is being used?
The Wizard's level 1 Burning Hands is a good example of an attack power that deals typed damage. This particular power uses fire damage:
Hit: 2d6 + Intelligence modifier fire damage.
Likewise, as an example for monsters, the Firelasher from Monster Manual 1 (page 104) has an attack that deals Fire damage. (You might notice the power itself also has the Fire type, but that's something separate from damage types.)
(Melee basic) Fire Lash (standard; at-will) ✦ Fire
Reach 2; +14 vs. Reflex, 2d8 + 5 fire damage.
The Fighter's level 1 Reaping Strike, on the other hand, deals damage of no type:
Hit: 1[W] + Strength modifier damage.
And the same goes for the Earthwind Ravager monster's Slam attack (from the same page as the Firelasher):
(Melee basic) Slam (standard; at-will)
Reach 2; +26 vs. Fortitude; 2d8 + 10 damage.
The reason there's no straightforward way to ignore melee attacks is because they, like all powers, come with all kinds of damage types. For instance, consider the Firelasher's melee attack which I just quoted. As far as damage types are concerned, it's just a fire attack, and whether it's melee or not is neither here nor there!
Player classes also have a range of damage types available among their powers: the Cleric has many melee, ranged and AOE powers which deal radiant damage, whereas the Fighter has many powers that deal damage of no type.
But the bracers' description say they can protect from virtually anything!
These clear crystal vambraces of interlocking plates can protect you from virtually anything, for a time. — Diamond Bracers' description
This holds no actual mechanical significance - it's exaggerating a bit, and no doubt the marketing pitch of a wiley equipment dealer. I'll quote Reading a Magic Item, from page 224 of the Player's Handbook (a similar paragraph appears in the Rules Compendium on page 279):
[The description entry] gives a brief description of the item, sometimes explaining what it does in plain language, other times offering flavorful information about its appearance, origin, effect or place in the world. This material isn't rules text; when you need to know the exact effect, look at the rules text below.
The Player's Handbook 1, on page 55, says nearly the same thing about the flavour text of Powers.
This would be a bit of a surprise to players from previous editions, like D&D 3.5e. In editions prior to 4e, the description was of supreme mechanical importance.
By contrast, 4e's philosophy is that descriptions don't matter mechanically and are just there to help you understand what the heck the power is supposed to do. Descriptions and flavour text themselves are entirely replaceable according to your own preference.
So I really can't resist untyped damage?
Not with this item, at least. Nothing - or very little - resists untyped damage, and that is its strength, in exchange for the fact that nothing is especially vulnerable to it either.
Untyped damage is D&D 4e's safe haven for damage dealing. Since nothing resists it, untyped damage you deal will always reliably land in a straightforward manner, even when your foe would resist all other damage types your available powers can deal.