"The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution, the Laws of the United States, and Treaties made, or which shall be made, under their Authority… In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction. In all the other Cases before mentioned, the supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make."
-Article III, U.S. Constitution
Some opponents of judicial review claim that it is unconstitutional based on the Tenth Amendment. This argument hinges on the recognition that judicial review is not an explicit power granted in the Constitution, but is inferred from other provisions of the Constitution. Since the Tenth Amendment reserves all powers that are not explicitly delegated to the federal government to the states, judicial review should be unconstitutional since it is not explicitly delegated. That is the basic argument of opponents.
The writers of the Constitution also had their disagreements over judicial review. Alexander Hamilton, for example, argued in favor of judicial review believing that allowing legislators to be "constitutional judges of their own powers" would allow them to substitute their own will over the will of the people. Hamilton saw the court as the intermediary between the people and the government "to keep the latter within the limits assigned to their authority."
Opponents of judicial review generally express the concern that judges will be swayed by their own beliefs and not driven by any firm rules. Thomas Jefferson believed that making judges "the ultimate arbiters of all constitutional questions" would lead to "the despotism of an oligarchy." He pointed out that judges are just as fallible as other men and more dangerous than other government officials because they are not elected and serve for life. Abraham Lincoln shared similar opinions believing that allowing the Court to pronounce "irrevocable" decisions, "the people will have ceased to be their own rulers."
Judicial Review or Judicial Activism
Application of judicial review often leads to charges of judicial activism. This is a political term that often changes meaning depending on who is making the charge, but is generally used to claim the judicial branch is behaving as legislators and writing new law. Judicial activism can be used when upholding or striking down laws, and simply overturning prior precedent is not by itself judicial activism. It's possible to support judicial review as well as oppose judicial activism.
In the past I've provided my own definition of judicial activism as going against the plain language of the Constitution in a) supporting a law or b) overturning a law, or creating new law purely through interpretation of the Constitution. I would argue that judicial review can certainly cross into the realm of activism, but it doesn't have to. Provided the Court avoids creating new laws and can justify its ruling in supporting or overturning a law based on the plain language of the Constitution, it is not engaging in judicial activism.
My Understanding of Judicial Review
Given all of the above, I think that judicial review is supportable through the Constitution and is clearly implied in Article III. That said, let me address some of the specific arguments mentioned earlier.
Irrevocable Decisions Lincoln opposed judicial review (after losing a case) believing that the people were losing power to rulers who issue irrevocable decisions. This is simply false. The Court's rulings are never irrevocable and the very process of judicial review permits the overturning of bad decisions at any point in the future.
Interpreting Constitutionality Jefferson described the Court as the ultimate arbiters of constitutionality, but we should expect the president and congress to make their own determinations on constitutionality. As such, all three branches are arbiters, and each is subject to review (and rebuke) by the other two.
Overruling the Court In addition, the legislative and executive branches have the ability to amen the Constitution and overrule the Court. This is a very high hurdle, but it requires agreement by both branches as well as voters and the states. But saying that it is difficult is not to say that it is impossible to overrule the Court if they get it wrong.
Ruling Based on Belief Lastly, the justices on the Supreme Court are not without rules, although critics from left and right often claim that they base their decisions on personal beliefs rather than law. In fact, one of the most important rules the justices follow is that laws are "presumptively valid," which is particularly important in the role of judicial reviewer.
In short, I support the idea of judicial review even though, as a conservative, the Court sometimes gets things wrong from my point of view. In a free country, I would expect it to be wrong sometimes, but I think the risk of losing is more acceptable than allowing congress or the president to determine for themselves what is or is not constitutional. Each branch of government has that responsibility, and each has the ability to check the others.