What keeps the resting potential of neurons constant at -70 mV?
The neuronal cell membrane is quite permeable to K+. Because the Na+,K+-ATPase pumps K+ inside of the cell, K+ tends to diffuse outward again, thereby taking positive charge outside the cell and making it negative inside (see Further Reading 1). This outward flow of K+ stops at a certain point, because the driving force of K+ diffusion out of the cell along with its concentration gradient, equals the charge gradient, which becomes more and more negative inside the cell as more K+ diffuses out (see Further Reading 2). Hence an equilibrium is reached which is close to the resting membrane potential of -70 mV. Other ions such as Cl- are also relatively permeable and affect the resting membrane potential (see Further Reading 1). Ions such as Na+ and Ca2+ are typically highly impermeable and do not substantially affect the resting potential. 

As to your second sub-question:voltage-gated channels are gated through voltage differences (Purves et al., 2001), not ion concentration differences. 

Further Readings
1. If the average resting potential of a neuron is -70 mV, why is there such a high ratio of K+ ions inside relative to out?
2. Why do negative ions flow into a cell in an inhibitory synapse, even though a neuron has a negative resting potential?   

Reference
-  Purves et al., Neuroscience (2001). 2nd ed. Sunderland (MA): Sinauer Associates