Not exactly. With only a few exceptions, the USA (and its western allies) were able to use military force or the threat of force to extert influence on a global basis, even in the face of Soviet condemnation. Korea, Vietnam, Malaysia, Lebanon, Syria, Suez, the Domincan Republic, El Salvador, Nicaragua, Grenada, Panama, former French West Africa, the Falklands, etc. The fact that the US or its allies may have not always succeeded speaks more to local resistance and domestic opposition than anything the USSR did.
The USA and its allies were able to place nuclear arms and other powerful forces in client states and allies bordering eastern Europe that were a direct threat to the interests of the USSR, and the Soviets almost found themselves in a nuclear war (one they would have resoundingly lost, by the way) when they tried the same thing in Cuba. The only place the USA or its allies would not exert their power was in eastern Europe, which they defacto recognized as off limits.
I'm not arguing that things didn't became substantially easier for the USA after the collapse of the USSR, but the situation was always far more assymetric than you would imagine.