Is it accurate to say that American common law is based on English common law, which is based on Roman common law?
American law is based on English common law. This much is true. In fact, most US states have something called a "reception statute," which explicitly states that the laws of England apply except where they conflict with a later enactment by the state legislature or the state constitution. For example, from the 1776 Delaware Constitution: "The common law of England, as-well as so much of the statute law as has been heretofore adopted in practice in this State, shall remain in force, unless they shall be altered by a future law of the legislature; such parts only excepted as are repugnant to the rights and privileges contained in this constitution, and the declaration of rights, &c., agreed to by this convention." New York's 1777 constitution has a similar clause. Alexander Hamilton argues in Federalist 84, that English liberties still applied under the newly-drafted Constitution, and would be protected. Even states that were admitted long after the Revolution include similar clauses in their law codes. Even today, California's Civil Code says, "The common law of England, so far as it is not repugnant to or inconsistent with the Constitution of the United States, or the Constitution or laws of this State, is the rule of decision in all the courts of this State."
That said, I think you're mistaken to say that the English common law is based on Roman common law. Quite the opposite. The laws of medieval England had were largely Germanic in origin, and the influence of Roman law was limited mostly to the Church in that era. Now, that's not to say that English lawyers didn't borrow Roman concepts, or that English law wasn't influenced by the Code of Justinian over the centuries. Far from it. (Codification of the law is a concept that was borrowed from the Romans, for instance.) The chancery courts and the admiralty courts of centuries past, for instance, were theoretically unrestricted by the law courts' reliance on precedent, and they borrow much more from Roman law, and even an authority like Blackstone uses maxims of the law from Justinian. But the foundation of the legal system has always been the Germanic common law.
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