The only way for the federal government to actually codify the results of the Roe v. Wade decision would be a Constitutional Amendment. Given that 2 in 3 Americans don't agree that abortion should be legal in all circumstances, then it should be needless to say that the requirement of 2/3 of both houses of Congress proposing it and 3/4 of states ratifying it is simply not going to happen there.
However, Congress could pass a law that simply bans any restriction of abortion by the states. This would require only the normal process for passing a law, i.e. majority vote of both houses of Congress and being signed by the President, though, in practice, a 60% supermajority might be required in the Senate to overcome a filibuster. Note, however, that this would not have anywhere near the legal effect of Roe or Casey, as it could be reversed just as easily as it was passed after a change of Congress and/or the administration. Congress cannot create Constitutional rights by passing a regular bill; only amending the Constitution can do that.
The Supremacy Clause of the U.S. Constitution (Article VI, paragraph 2) states that (emphasis mine),
This Constitution, and the laws of the United States which shall be made in pursuance thereof; and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land; and the judges in every state shall be bound thereby, anything in the Constitution or laws of any State to the contrary notwithstanding.
In other words, any law that Congress makes under its authority granted in the Constitution preempts any state law (including provisions of state Constitutions) to the contrary. So, if Congress passed a (legal) law stating that states can't place further restrictions on abortion, then they can't, as long as that law stands.
The legal basis for making such a law would most likely be simply the Commerce Clause, which they already use to justify just about anything Congress does. The Commerce Clause is paragraph 3 of Article I, Section 8 of the U.S. Constitution, which reads (emphasis mine),
Congress shall have power...
To regulate commerce with foreign nations, and among the several states, and with the Indian tribes;
Courts have allowed Congress to get away with pretty much anything that they can argue "affects interstate commerce" under the Commerce Clause, even if the actual actions being regulated occur entirely within a state in a given case. The relation is often extremely tenuous and this would probably be one of the less egregious examples if it were passed. States would probably challenge it in court anyway (if it didn't get repealed before they had time to do so,) but it's likely that the challenges would not succeed, at least not on the argument that it wasn't authorized under the Commerce Clause.
Granted, this is assuming that current court precedent on the Commerce Clause stands and the Supreme Court does not choose to narrow its interpretation of the clause to one requiring more direct relation to actual interstate commerce. Given the change in the makeup of the court over the past several years, it is possible that the court could narrow that interpretation in a future decision.