No,they are both 30 caliber.
the Ruskie is rimmed the NATO is rimless and the NATO pressure is WAY higher.
They are loaded to similiar pressure levels. The 762x54R is actually between the 762 Nato and the old 30/06 is power. The NATO round is a 762x51,and IF I remember correctly,the 30/06 is a 7.62x63.
Having said all this,why bother? It's really not worth the expense,especially when there is plenty of 762x54R ammo around,and it is slightly more powerful than the 7.62 NATO to begin with.
If you absolutly,positively have to have a .308 Winchester rifle,buy a Savage bolt action with the heavy barrell. Do NOT buy one of the converted M95 or M96 Mausers with only 2 locking lugs. The old 7X57 Mauser round they were made for IS loaded to a much lower pressure than current 7.62 NATO ammo,and the rifles are unsafe when altered.