How does Newtonian Gravity explain the difference between the experience of 1G and zero gravity? Or can it? Humans can feel the difference between


*

*being on earth's ground, and

*being in space (zero gravity)


This can be explained by general relativity, where in (1), force is applied by the ground upwards, and in (2) there's no force.
How does Newtonian Gravity explain the phenomenon? In both cases, there's zero total force applied and subject is not moving.
If Newtonian Gravity couldn't explain it, how come no one thought about it at the time? Is it because no one had been away far enough from earth's gravitational field in order to experience zero gravity?
 A: In newtonian mechanics gravity is a so called body force, that acts throughout the volume of a body. Since a body force produces the same acceleration for every small mass element of a body, it doesn't create internal mechanical stress between parts of the body. 
(Well, for extended bodies gravity can create tidal forces, but they are very small for everyday objects. Tidal forces only exist in an inhomogeneous gravitational field.)
In contrast to that, the normal force from the ground that holds you up against gravity is a surface force, that acts across an internal or external surface element in a material body. Surface forces create internal mechanical stress inside a body. While body forces can directly act on any mass element inside a body, surface forces have to "travel" from one mass element to the next. 
For example when standing on the ground the normal force pushes against you feet, your feet push against your leg, you leg against your hip and so on. You can feel a difference when the surface force acts on a different part of your body. While standing you feel pressure on your feet, but while sitting you feel the pressure on your buttocks.
The deformation of the corresponding sensory cells due to mechanical stress gives the perception of a force. Therefore you don't percieve any force, when only gravity is acting on you while in free fall.
A: Neither Newtonian gravity nor General Relativity by themselves can explain the different sensations on the earth or in orbit. Both are theories of gravity and do not explain biological sensations. However, once you have added in the pressure sensors in the feet and the accelerometers in the inner ear, then both theories can explain the sensations as appropriate activations of the sensors. 
In Newtonian gravity the pressure sensors are activated as the contact force from the ground pushes up on the feet in 1 g, and are not activated in 0 g since there is no contact force. Similarly, in 1 g the fluid in the inner ear pools toward the bottom of the canals, while in 0 g the fluid is not pulled towards the bottom. 
A: Newtonian Mechanics states that there is a force $F_g = G \frac{m_1 m_2}{r^2}$ being applied on a human in free fall. How did we know it was real at the time? Lets say the human has a mass $m_1$, 
then in the Newtonian world the way we know this force is real is that if we substitute $m_2$ with Earth's mass then the quantity $ G \frac{m_2}{r^2} $ is actually equal to the acceleration of the human relative to the center of the Earth. 
Now when a human is on earth as you noted in order for them not to "fall through the soil" the soil must be solid and exerting a normal force upwards equal to the force of gravity. 
So what exactly is this feeling of "weightlessness"? It is the absence of the "normal force" of soil pushing against our feet (or when we are not physically up to it, then our whole body). 
From this point of a view a free falling observer and a someone in a distant near 0-gravity region of space would more or less feel the same thing... "weightlessness" as both are not experiencing the normal force of land pushing them back. 
The normal force is a concept that predates general relativity (about as old as Newtonian mechanics itself if I had to guess) and it explains what "weightlessness" really means and why someone experiencing no gravitational force, or someone free-falling both feel equally "weightless". 
A: It was an empirical observation, not a theoretical prediction
The reason that gravity feels different when standing on Earth vs when in free fall (ie “space”) is because of something known as the Galilean equivalence (also known as weak equivalence) principle, which states that the gravitational mass is equal to an objects inertial mass. Or more commonly:  all objects fall at the same acceleration.
The fact that all mass appears to also have gravitational attraction is not something that was proven from Newton’s theories. Newton’s laws of motion and theory of gravity are agnostic on this fact. The equivalence had to be shown through experiment, most famously by Loránd Eötvös.
General Relativity sidesteps the equivalence problem by taking it as an axiom; it is simply assumed true. 
