What is the outcome when a request for a writ of certiorari is denied and the court refuses to hear the case?

If the U.S. Supreme Court denies a petition for a writ of certiorari (a request to hear a case on appeal), then the decision of the lower court is final.
Denial of certiorari occurs in 98-99% of cases, and in no way implies that the
court agrees with the lower court's decision. Denial only means that the case,
as presented, isn't of sufficient importance to warrant a review, doesn't
involve constitutional issues, conforms to a precedent already set, falls
outside the court's jurisdiction, or is moot.