Yes, it is a uniquely American development created during the war in Viet Nam.
(background info) A fire support base is sort of mobile artillery encampment. It is set up to provide temporary fire power in a local area where fighting (usually of an insurgency character) is occurring. It differs from normal warfare in that there is no front. The FSB is presumably surrounded by the enemy or by places from which the enemy could operate.
The applicability to Viet Nam is obvious: in Viet Nam, the VC used tunnels and secret supply routes to fight in wide ranging areas, conducting a guerilla war which could flare up into significant local conflicts involving artillery. The FSB was a response to this, allowing the US Army to place fire power in hotspots where NVA activity was occurring.
The general concept of a strong point is, of course, not new. Roman legions operated in a similar way, setting up fortified camps deep in enemy territory and operating out of those camps. Also, to take an example from medieval times, after the Normans invaded Britain they were faced with trying to control a large, hostile country. They set up castles, strongpoints, throughout the country from which they could exert control and launch military operations.
FSBs are a little different in that they are artillery focused, the infantry is usually based elsewhere in many different smaller camps. The centralized FSB allows a single artillery center to support numerous smaller infantry strong points scattered around. This is possible because a 155mm gun can easily fire 10-12 miles away, so a single FSB can be used across a wide area.