I'd say yes, they do, based primarily on the following:
"Armed" Unarmed Attacks: Sometimes a character's or creature's unarmed
attack counts as an armed attack. A monk, a character with the
Improved Unarmed Strike feat, a spellcaster delivering a touch attack
spell, and a creature with natural physical weapons all count as being
armed (see natural attacks).
– d20pfsrd (emphasis mine)
Also, if natural weapons didn't count as melee weapons, creatures would be unable to deliver a coup de grace, since its rules explicitly say:
As a full-round action, you can use a melee weapon to deliver a coup
de grace (pronounced "coo day grahs") to a helpless opponent. You can also use a bow or crossbow, provided you are adjacent to the target.
– d20pfsrd (emphasis mine)
So, if a natural weapon isn't a melee weapon, then you could deliver a coup de grace with a dagger, but a dragon couldn't deliver one with a foot-long claw. That would be weird. :)
Sure, whether natural weapons are melee weapons isn't explicitly stated in the RAW, but it seems strongly implied... to me, at least. (Check also that some creatures with natural weapons only have Weapon Focus as a feat, etc.)