If the Supreme Court decides not to hear a case what then becomes the final decision in that case?
The decision of the previous appeals court that heard the case is the final decision should the Supreme Court refuse to hear the case.
The Supreme Court decides cases that are appealed by a lower court; a lower court has made a decision and one of the parties feels strongly enough that the decision was wrong that they make an appeal to the Supreme Court. The Supreme Court reviews the cases and determines which ones they will hear, they have the ability to decline to review a case. The Supreme Court doesn't hear only appeals, there are situations where…
Since it is the Supreme Court that decides what is constitutional and what is not, the decisions of the Supreme Court cannot be unconstitutional, however, it is always possible for the Supreme Court to make new decisions which reverse older decisions. So in theory, if the Supreme Court does something wrong, they will be reversed by a later sitting of the same court (but with new judges).