You are confusing a few concepts.
One is the distinction between what are known as "common law" jurisdictions derived from the English legal system, and "civil law" jurisdictions derived from one of the continental European legal systems that is ultimately derived from Roman law.
Another is the distinction between determining the meaning of ambiguous legislation, which all courts do by definition, and the power of judicial review, which overturns legislation which is invalid for some reason rather than merely trying to interpret an ambiguous provision.
Ambiguous means "unclear" or "capable of being interpreted in more than one way" and every time every court encounters unclear legislation it must decide what it means, even if it is not invalidated.
In contrast, some judiciaries that have the power of judicial review and those that do not. Judiciaries that can declare a law to be invalid have the power of judicial review. Judiciaries that cannot declare a law to be invalid do not have the power of judicial review. Every state and federal U.S. Court at every level (not just the U.S. Supreme Court) has the power and obligation to declare that a law violated the U.S. Constitution. In many countries, no court, or only a "constitutional court" has the power to make declare legislation to be invalid by exercising judicial review.
Every time that a legislature passes a statute on a subject covered by common law (i.e. judge-made law derived from case decisions that serve as precedents), it shrinks the scope of common law relative to statutes. And, in principle, almost all of the common law could be replaced by statutes without all that much difficulty. But, in civil law countries, statutes are frequently comprehensive and are the sole source of legal authority about their subject matter superseding all case law, while in common law countries, statutes are often piecemeal tweaks to a common law background that is assumed by the statute.
For example, every civil law country would have a comprehensive statute setting forth the principles of contract law, while a typical common law jurisdiction might have a statute that declares that certain contracts must be in writing but does not comprehensively set forth the law of contracts in all circumstances.
There are some features of civil law countries, such as the absence of jury trials, which cannot be constitutionally changed to the civil law system, even in jurisdictions such a Puerto Rico and Louisiana in the United States which have civil law roots prior to joining the U.S. (at least in criminal cases and in the federal courts).
The power of judicial review (i.e. the power of courts to declare a statute unconstitutional and void) is also inherent in the U.S. Constitutional system of government and could not be removed without a constitutional amendment. There are common law countries, e.g., England, which did not historically have the power of judicial review, which was an innovation for a common law countries such as the United States when it was first invoked. (For what it is worth, India goes one step further; its Supreme Court asserts and exercises the right to declare portions of its own constitution to be unconstitutional.)
There are other aspects of civil law legal systems which would probably also be declared unconstitutional in the United States as well, such as the lack of a prohibition on the introduction of hearsay evidence in criminal trials which violates a provision of the U.S. Constitution's Bill of Rights known as the "confrontation clause."
It is unclear to me whether the principle that case law precedents have binding legal effect in future cases, which is part of the common law system that is absent in the civil law system, has a constitutional dimension or could be displaced by law.
But, most aspects of a civil law legal system could be adopted in the United States if the relevant legislatures so desired. Indeed, many aspects of the U.S. legal system have moved in that direction. For example, only a handful of U.S. states now recognize the concept of a "common law crime". Almost all states now only allow criminal sanctions for crimes codified by statute, which was not the case at the time of the American Revolution, when few crimes were codified.
Obviously, with a constitutional amendment, almost any change to the U.S. legal system is possible.