NO, the Supreme Court isn’t the only branch tasked with determining the constitutionality of “laws”

Here’s Ronald Reagan’s Attorney General on the subject:

“Constitutional interpretation is not the business of the Court only, but also properly the business of all branches of government.

The Supreme Court, then, is not the only interpreter of the Constitution. Each of the three coordinate branches of government created and empowered by the Constitution-the executive and legislative no less than the judicial-has a duty to interpret the Constitution in the performance of its official functions. In fact, every official takes an oath precisely to that effect.”
–Edwin Meese

Meese added that in order for the United States to be a nation of laws, the constitution must come first, not just any law passed by a congress and signed by a president. Law is, of course, only that which is constitutional.

He wrote:

If law, as Thomas Paine once said, is to remain “King” in America we must insist that every department of our government,every official, and every citizen be bound by the Constitution. That’s what it means to be “a nation of laws, not of men.”

Michael Boldin