Usually not. In most states, the court system is made up of trial courts, who are reviewed by appellate courts, who can be reviewed by the highest court, which is usually called the Supreme Court. However, not all states call their highest court the Supreme Court. New York is an example.
In most states, a litigant who has lost a trial at the trial court level (decided by a judge or jury) has the right to appeal that decision to an intermediate court. There are specific deadlines and reasons which must be met in order to properly perfect such an appeal. There are also a couple of different types of appeals, one which suspends the execution of any judgment and one that does not.
Once an issue is decided by the intermediate appellate court, a litigant's options are limited. Except in death penalty cases, there is not an automatic right to an appeal of the appellate court's decision to the State Supreme Court. A litigant which seeks review by the Supreme Court may apply for what is called a Writ of Certiori. This Writ is a brief wherein the litigant asks the court to hear the case. The State Supreme Court has specific rules that it publishes that establishes the types of cases and issues which a litigant can ask them to review. If the issues involved in the case do not fall in that category, then the litigant may not seek a Writ application to that court.
Even if the issues fit into a category that allows the litigant to make a Writ application to the Supreme Court, most Writ applications are denied without any explanation being provided by the court. Under the Supreme Court rules, a denial of a writ application has no precedential value and makes no statement about the court's opinion of the issues presented. The court has simply refused to hear arguments for unknown reasons.
A litigant has only a small window of time to even seek a Writ application to the State Supreme Court following the decision by the appellate court. Since the Supreme Court receives so many Writ applications each year, it must be careful to only accept those few cases where the law is unclear or where there is a split of opinion between several of the lower courts. Thus, the Supreme Court usually only accepts cases that will have far-reaching effects on future cases.