Yes it will fire 7.62 NATO, or 7.62x51 (same thing). I have ran both through my ruger and had no issues. The Remy should be no different. I did however have better accuracy results with the .308win cartridges. In factory ammo, Federal Gold Medal Match is hard to beat. Mine likes the 168gr best. YMMV.
The headspacing is different between a 7.62x51 NATO and .308 WIN, the NATO chamber is quite a bit longer. So the problem is shooting commercial .308 WIN through a NATO chamber and the shoulder / brass being blown out beyond the norm.
In your case, so long as the round chambers ... then you are fine.
The military ammo is not very accurate, especially out of barrels with faster rates of twist (1:11 - 1:9) like a Remington 700 5R, or an NM M1A.