Why are resistor values constant in the face of changing voltage and current?
When the resistance value is constant (doesn't depend on I or V), we call that a linear resistor. 

Common resistive materials (such as metals, carbon, etc.) are reasonably linear to a first order approximation. There is no physical mechanism that strongly causes them to "push back more" when a greater voltage is applied.

If you look more carefully, you will usually find that as you increase the current through a real resistor it heats itself up, and this does cause the resistance to change. However we try to choose materials for our resistors where this effect is small. TCR (temperature coefficient of resistance) values on the order of 100 or 200 ppm/C are easily available.

A diode (in a dc circuit) is an example of what we call a nonlinear resistor. A resistor whose resistance depends on the current through it. However, these tend to resist less, the more current they are passing.