In observing the falling apple, Newton realised the apple was accelerating towards the Earth, and in conjunction with his 2nd law, he knew there must be a force acting on this apple to cause the acceleration. This he called gravity and the associated acceleration "acceleration due to gravity". If you imagine the apple tree to be twice as tall, we expect the apple to still be accelerated toward the ground, hence meaning he concluded that gravity must extend a long way, and for example affect the moon.
By thinking through these stages (and a few other ones), he concluded that any two objects in the universe exert gravitational attraction on each other (Newton's 3rd law: If object A exerts a force on object B, object B exerts an equal and opposite force on object A), and led him on to create the law of universal gravitation.
Depending on how complicated you want it, he basically "discovered" gravity from it, and created the above law from it. He also created the Gravitational constant G which is thought to be universal (same in all places at all times), and hence universally characterises the intrinsic strength of the gravitational force.
