Although the Erie decision itself does not identify specific provisions of the Constitution violated by Swift, the language of the decision implies that Swift had stolen powers reserved to the states, in violation of the Tenth Amendment. Justice Brandeis also noted problems for equal protection of the laws, but the Equal Protection Clause of the Fourteenth Amendment applies only to states, and the Fifth Amendment's Due Process Clause was not read to include an equal protection component until the 1954 decision in Bolling v. Sharpe.
As a result of Erie, each federal district court was required to apply the law of whichever state it was sitting in, as though it was a state court of that state. Of course, this was a very difficult decision for the Court, since overruling Swift meant that a huge number of decisions by the Court and all lower federal courts were no longer valid law.
However, the Court did not declare the Rules of Decision Act itself unconstitutional. Instead, it reinterpreted the Act so federal district courts hearing cases in diversity jurisdiction had to apply the entire law, both statutory and judge-made, of the states in which they sit.
This post was edited on 10/31 at 1:49 pm