Can a dispute decided by an alternative Dispute Resolution method be enforced if the losing party refuses to go through with the decision? if so how?

It depends on what form of ADR was used. Every state (and the federal government) has a statute for the enforcement of arbitration awards. If the ADR was a different form than arbitration, the settlement can still be enforced, provided that it can be proven. The enforcement method in every case involves commencing a lawsuit and alleging the outcome of the settlement process as the cause of action.