Short answer: Yes, you can fire common 7.62X51 military ammo in any currently produced rifle chambered for .308 Winchester.
The 7.62x51 NATO cartridge is built to certain military specifications. The round must be able to operate semi-automatic rifles and belt-fed machine guns. The current specifications for 7.62x51 limit it to a pressure of 50,000 psi to operate the M14 and other semi-auto gas guns, along with those machine guns that the military uses.
Currently produced .308 Winchester ammunition is SAAMI standardized at 62.000 psi. There are also small difference in cartridge specs between the military round and the .308 Win as currently produced, but the basic understanding is that the military cartridge will fit and fire in currently produced .308 rifles.
The .308 Winchester is a very versatile cartridge. Learn more about it here
But yes, you can fire NATO ammo in your rifle. We use a lot of NATO ammo as practice ammo for our precision rifle team. If you can find some Lake City M118 LR ammunition at a reasonable price, buy all you can afford. That particular ammo is considered very good ammunition for the .308 Winchester.