I believe the US is one of the largest exporter of guns in the world, so yes, America is "filling the world" with guns. Though this sounds a bit melodramatic, it still may be true. As long as their are buyers for guns, there will be a market. If that market didn't come from America, it would come from somewhere else.
America can produce many weapons or any number of items for which there is a demand. There is a demand for guns worldwide, and there always will be. America isn't producing them; provide citizens are. If America wasn't producing them, other countries would fill the void. Blaming America will have no impact on the worldwide supply of guns.
The United States is not to blame for illegal guns flowing around the world. We are not the only nation to produce firearms or have crime problems. As long as guns are produced and greedy people are willing to be bribed illegal guns will continue to flow through the black market.
No, America is not necessarily filling the world up with guns. While a large exporter of arms, there are many household names in the weapons industry that are based in various countries. For instance, Heckler and Koch is a German company while Beretta is an Italian arms manufacturer. Since these companies have existed for many years and account for a large volume of gun sales, America is not the sole problem country for filling the world with guns.