It is, of course, impossible to know this. But my opinion is that the United States could not have continued to exist if the Anti-Federalists had won. This is because the US would have looked too much like it had under the Articles of Confederation and would have eventually broken up.
Under the Articles of Confederation, the states had way too much power and the federal government had too little. The US looked more like the United Nations does today with the central government unable to control the states and the states being in conflict with one another quite often. I think that if the Anti-Federalists had won, this sort of dynamic would have continued. There would have been little to bind the states together than they would have split up into smaller and separate nations.