No. It has authority over the original jurisdiction of the Federal District Courts and the Appellate jurisdiction of the Federal Circuit Courts of Appeals. It also has jurisdiction over the Appellate--but not the Constitutionally delegated jurisdiction of the Federal Supreme Court. It could, any time it wanted, as I have pointed out, stop the Federal Courts from overturning State efforts to stop abortion on demand by a simple act, taking away the jurisdiction to hear such cases.
But it does not have such jurisdiction over the State Courts. And if you read the Declaration of Independence, you will see that the right to a local judiciary, independent of interference by a remote central authority, was one of the principles for which they fought.
Section 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
Section 5. The Congress shall have power to enforce, by appropriate legislation, the provisions of this article.
Congress believed this to be the case. Otherwise they would not have acted.