No. Unarmed strikes are not natural weapon attacks.
When a creature that can use manufactured weapons does not have a weapon, they can make an unarmed attack. This attack is usually weaker than using a manufactured weapon, but there are exceptions such as the monk or brawler classes. For full-round attacks, unarmed strikes use iterative attack rolls (BAB, BAB-5, BAB-10, etc). Also, an unarmed character making an unarmed attack against an armed opponent is risky:
Attacks of Opportunity: Attacking unarmed provokes an attack of opportunity from the character you attack, provided she is armed. The attack of opportunity comes before your attack. An unarmed attack does not provoke attacks of opportunity from other foes, nor does it provoke an attack of opportunity from an unarmed foe.
An unarmed character can’t take attacks of opportunity (but see “Armed” Unarmed Attacks, below).
I suggest reading through the combat rules about unarmed attacks for more detail.
A natural attack (or natural weapon) is something that most humanoid player races don't have. It is part of their body, such as a claw or bite, or maybe just a clumsy slam or tail slap. Creatures are generally proficient with their own natural weapons.
As an alternative to making a full-round attack of iterative unarmed rolls (i.e., creatures with natural weapons can make unarmed attacks, but they generally don't), a creature can instead make a full-round attack with only their natural weapons. Natural attacks also don't use iterative rolls in the way that unarmed attacks do:
Most creatures possess one or more natural attacks (attacks made without a weapon). These attacks fall into one of two categories, primary and secondary attacks. Primary attacks are made using the creature’s full base attack bonus and add the creature’s full Strength bonus on damage rolls. Secondary attacks are made using the creature’s base attack bonus –5 and add only 1/2 the creature’s Strength bonus on damage rolls. If a creature has only one natural attack, it is always made using the creature’s full base attack bonus and adds 1-1/2 times the creature’s Strength bonus on damage rolls. This increase does not apply if the creature has multiple attacks but only takes one.
Creatures with natural weapons also have the option to make a full-round attacks with both their manufactured weapons (or unarmed attacks) in addition to unoccupied natural weapons. However, all of the natural attacks are considered secondary during these attacks, which means sacrificing accuracy and damage.