No, we shouldn't. The UN Treaty you reference doesn't even exist yet, for one thing. No formal proposal has been submitted yet. So there's one reason not to fear.
Second, this treaty, as it has been discussed, deals strictly with the international trade. It says nothing and never has been pointed towards domestic arms sales. Furthermore, the treaty does not ban the sale of arms or the private ownership of guns. The theoretical treaty is strictly concerned with preventing the sale of arms to terrorists, organized crime, and those who wish to commit genocide.
There is a separate treaty that would like to take a stab at domestic arms sales, but unlike the Small Arms Trade Treaty, this other treaty has not received any recognition or support from the United States. Even if this treaty were signed by the President, it would not be considered binding on the USA unless it was ratified by a 2/3 majority vote in the Senate. And even if that happened, the Constitution trumps any and all international treaties, as ruled in Reid vs. Covert.