That's not true at all. The United States would have continued among the states remaining in the federal union. When a state secedes from a federal union, it is not the case that the union between all of the states is dissolved; only that union between the seceding state and the other states in the union is dissolved.
A federal union is somewhat like a maximally connected graph, with states as the nodes, and the federal union consisting of the edges. When a state (node) leaves the federal union, it severs all of its immediate connections. The other connections among the other nodes, however, remain intact.
It seems that there has historically been much confusion on this matter, extending back even to contemporary events in the 1860s. Some people seemed to think then, and perhaps some still think now, that a dissolution of the union between a state and several other states means that the entire union is dissolved. But as we have seen, this is certainly not the case.