Democrats in Congress have mentioned codifying Roe v. Wade into law (whatever that means). However, I was under the impression that Congress couldn't overturn a Supreme Court decision on a Constitutional question (in this case, the question of abortion rights in the US). The overturning Dobbs decision, therefore, remains intact and there's nothing Congress can do, short of "packing the court."
The question reads:
I was under the impression that Congress couldn't overturn a Supreme Court decision on a Constitutional question
That is, in general, correct.
The overturning Dobbs decision, therefore, remains intact and there's nothing Congress can do, short of "packing the court."
That does not follow. There are a number of steps that Congress could take, if it chose to, that would have the effect of providing abortion access to people in the US, in states whose legislatures have voted to restrict or ban it. Whether Congress will take any of these actions is not clear, that is a political question.
Passage of an Enforcement Law
Section 5 of the 14th Amendment reads:
The Congress shall have the power to enforce, by appropriate legislation, the provisions of this article.
The scope of power that Congress has under this provision is wide, and has been little used. Congress could pass a law declaring that certain abortion restrictions violate the rights granted under the amendment, and prohibiting states from enforcing such restrictions. Such a law would be subject to court challenge, and whether it would be held to be a valid exercise of the power of Congress is uncertain. But Congress could try this route.
Passage of a Law Restricting the States on Other Grounds
Congress could pass a law regulating abortion, thereby preempting state laws on the subject that conflicted with the new federal law. Such a law would only be valid if it were within the powers granted to the Federal Government. There has been debate on which, if any, of those powers would permit such a law. Given the degree of regulation of medical issues already permitted under such laws as HIPPA, the ADA, and the acts establishing Medicare, and the extensive scope of Commerce Clause regulation during the past 60 years and more, such a law might be held to be within the powers of Congress.
Providing Access Within Federal Facilities
Article I, section 8 of the US Constitution reads, in pertinent part:
Congress shall have the power...
To exercise exclusive Legislation in all Cases whatsoever, over such District (not exceeding ten Miles square) as may, by Cession of particular States, and the Acceptance of Congress, become the Seat of the Government of the United States, and to exercise like Authority over all Places purchased by the Consent of the Legislature of the State in which the Same shall be, for the Erection of Forts, Magazines, Arsenals, dock-Yards, and other needful Buildings [emphasis added]
This means that Congress may, by law, set the rules for all activities in federal facilities. Congress could establish abortion clinics within federal facilities such as military bases, national parks, or federal office buildings, if it decided to. Under the Supremacy Clause, such laws would override any conflicting state laws.
Article II section 2 provides in pertinent part:
In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction. In all the other Cases before mentioned, the supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make. [emphasis added]
Congress can, under this provision, remove categories of cases from the reach of the Supreme Court. This is known as "jurisdiction stripping" This has not often been done, but the power is clear, and has been upheld in the past, for example in the case of Ex parte McCardle, 74 U.S. (7 Wall.) 506 (1869). Such a law could be used to insulate laws discussed above from Supreme Court review.
It should be noted that all the above are things that Congress could do, or might be able to do, to extend abortion access in states that have passed laws banning or restricting abortion. Congress has not passed laws doing any of these things, and it may never do so. If Congress took any such actions, court challenges would be likely, and there is no telling in advance how such challenges would be resolved.
They can't overturn the decision; but they don't have to.
The holding in Dobbs was that the US Constitution does not provide a right to an abortion, and so a state law prohibiting abortions is not unconstitutional. Congress can't "overturn" it in the sense that they can't make the Constitution provide such a right, short of a constitutional amendment.
But that in itself doesn't stop them from providing such a right in other ways, e.g. through ordinary legislation. Under the Supremacy Clause, such a law would supersede any state bans; provided that it falls within the scope of Congress's enumerated powers, which assertion would itself probably be challenged in court. If it's within their legislative power, then there's no conflict with Dobbs.
To give a more mundane example, nobody thinks that there is a constitutional right to have an airline ticket refunded within 24 hours of purchase. If the Supreme Court ever had occasion to rule on the question, they would surely hold that nothing in the US Constitution says that people have this right. But Congress does have the power, under the Commerce Clause, to pass legislation that confers such a right on consumers. They have done so, and this law would not in any way conflict with the aforementioned hypothetical Supreme Court ruling.
A court has jurisdiction to impose judgments upon the parties to cases before it. Higher-court judges generally publish advice on the principles they will use when reaching their decisions, so that lower courts can reach the same decisions themselves directly, without every case having to be appealed to the higher court. If part of a statute is written in a manner which could be interpreted as favoring either party, a higher court will, as part of its decision making process, say how it interpreted the ambiguous parts of the statute. If the court interprets a statute in a manner consistent with what it was intended to say (or at least consistent with what those currently in the legislature would want it to say), the legislature can simply allow the legislation to remain as it is, despite the ambiguity. If the court interprets the statute in a manner which is reasonable, but which does not coincide with what the legislature wanted, then the legislature can change the law to better state their intention.
Suppose there is a "guess the number of the jellybeans in a jar" contest, and after all guesses are submitted, including guesses for 998, 999, and 1000, it is discovered that the jar contains 499 whole red jellybeans along with a red half-jellybean, and 499 whole blue jellybeans along with a blue half-jellybean. If there is nothing in the rules about how partial jellybeans should be counted, the entity responsible for judging the contest would have the authority to award the prize to either the person who said 998, the one who said 999, or the one who said 1000. The judge would be entitled to decide whether half jellybeans should be excluded from the count, should count as whole jellybeans, or should be counted numerically as half a jellybean with the overall result rounded to the nearest bean, and such decision would be final with respect to that particular contest outcome. On the other hand, no matter which counting method the judge chose, there would be nothing improper about the sponsors of the contest publishing rules announcing that in all of their future contests different rules would be applied.
The method by which Congress can overturn a SCOTUS decision is to amend the Constitution, which requires a good number of state legislatures to come to agreement. An amendment to recognize the right to Abortion could, in theory could be made by these methods. However, it's easier said than done. Similar amendments have been made (Notably, the 13th Amendment which banned Slavery save for hard labor punishments, the 18 amendment (prohibition). In the former two cases, it moved the ability to legislate on the issue from the states to the feds. In the later, it reversed this removal.
The 13th Amendment overturned numerous prior SCOTUS rulings because now that Congress had the ability to make national laws on slavery, it instantly overturned several regrettable decisions SCOTUS made on the issues.
The 14th Amendment was to patch a sort of loophole that existed because of the 9th and 10th amendment, forcing states to comply with the rights protected in the Federal Constitution (Without it, Congress might be limited by restrictions in the Bill of Rights, but the states were not so limited by the U.S. Constitution and states could make laws that violated the Bill of Rights).