Article the twelfth (AKA the 10th amendment) of the Bill of Rights from the US Constitution: "The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.”

So the current argument might be that since the constitution does not give the federal government the power to regular abortions (or to prevent states from regulating them), then that power to regulate them or not is reserved for the states to do as they please.

The prior argument (from Roe vs Wade) is that the US Constitution provides a right to privacy and that prohibiting or otherwise restricting abortions violates that right to privacy.

While there's no explicit right to privacy in the constitution, it could be implied by the 14th amendment: "... nor shall any State deprive any person of life, liberty, or property, without due process of law ..."

From the original Roe vs Wade opinion: "A person may choose to have an abortion until a fetus becomes viable, based on the right to privacy contained in the Due Process Clause of the Fourteenth Amendment. Viability means the ability to live outside the womb, which usually happens between 24 and 28 weeks after conception." https://supreme.justia.com/cases/federal/us/410/113/