The relevant portion of the Dobbs decision states:
We must now decide what standard will govern if state abortion
regulations undergo constitutional challenge and whether the law
before us satisfies the appropriate standard.
Under our precedents, rational-basis review is the appropriate
standard for such challenges. As we have explained, procuring an
abortion is not a fundamental constitutional right because such a
right has no basis in the Constitution’s text or in our Nation’s
history. See supra, at 8–39.
It follows that the States may regulate abortion for legitimate
reasons, and when such regulations are challenged under the
Constitution, courts cannot “substitute their social and economic
beliefs for the judgment of legislative bodies.” Ferguson, 372 U. S.,
at 729–730; see also Dandridge v. Williams, 397 U. S. 471, 484–486
(1970); United States v. Carolene Products Co., 304 U. S. 144, 152
(1938). That respect for a legislature’s judgment applies even when
the laws at issue concern matters of great social significance and
moral substance. See, e.g., Board of Trustees of Univ. of Ala. v.
Garrett, 531 U. S. 356, 365–368 (2001) (“treatment of the disabled”);
Glucksberg, 521 U. S., at 728 (“assisted suicide”); San Antonio
Independent School Dist. v. Rodriguez, 411 U. S. 1, 32–35, 55 (1973)
(“financing public education”).
A law regulating abortion, like other health and welfare laws, is
entitled to a “strong presumption of validity.” Heller v. Doe, 509 U.
S. 312, 319 (1993). It must be sustained if there is a rational basis
on which the legislature could have thought that it would serve
legitimate state interests. Id., at 320; FCC v. Beach Communications,
Inc., 508 U. S. 307, 313 (1993); New Orleans v. Dukes, 427 U. S. 297,
303 (1976) (per curiam); Williamson v. Lee Optical of Okla., Inc., 348
U. S. 483, 491 (1955). These legitimate interests include respect for
and preservation of prenatal life at all stages of development,
Gonzales, 550 U. S., at 157–158; the protection of maternal health and
safety; the elimination of particularly gruesome or barbaric medical
procedures; the preservation of the integrity of the medical
profession; the mitigation of fetal pain; and the prevention of
discrimination on the basis of race, sex, or disability. See id., at
156– 157; Roe, 410 U. S., at 150; cf. Glucksberg, 521 U. S., at 728–
731 (identifying similar interests).
These legitimate interests justify Mississippi’s Gestational Age Act.
Except “in a medical emergency or in the case of a severe fetal
abnormality,” the statute prohibits abortion “if the probable
gestational age of the unborn human being has been determined to be
greater than fifteen (15) weeks.” Miss. Code Ann. §41–41–191(4)(b).
The Mississippi Legislature’s findings recount the stages of “human
prenatal development” and assert the State’s interest in “protecting
the life of the unborn.” §2(b)(i). The legislature also found that
abortions performed after 15 weeks typically use the dilation and
evacuation procedure, and the legislature found the use of this
procedure “for nontherapeutic or elective reasons [to be] a barbaric
practice, dangerous for the maternal patient, and demeaning to the
medical profession.” §2(b)(i)(8); see also Gonzales, 550 U. S., at
135–143 (describing such procedures). These legitimate interests
provide a rational basis for the Gestational Age Act, and it follows
that respondents’ constitutional challenge must fail.
We end this opinion where we began. Abortion presents a profound
moral question. The Constitution does not prohibit the citizens of
each State from regulating or prohibiting abortion. Roe and Casey
arrogated that authority. We now overrule those decisions and return
that authority to the people and their elected representatives.
This clearly authorizes state legislation on the subject, and the earlier portion of the opinion clearly states that the U.S. Constitution does not give rise to a self-executing right to choose as a matter of substantive due process.
But, the Dobbs opinion does not really explore the boundaries of Congressional action to regulate abortion, under grounds such as the Commerce Clause, or the Enforcement authority of the 13th and 14th Amendments, although surely Dobbs contemplates that Congress could legislate with respect to territories of the United States that are not part of any U.S. state if it wished to do so in lieu of state law.
In modern U.S. history, the commerce clause authority of Congress has been almost unlimited, but as another answer notes, there has been conservative pushback against the general rule that Congress can enact any legislation, for example, in US v. Lopez, 514 US 549, and more recently, with respect to certain provisions of the Affordable Care Act (i.e. Obamacare).
It is hard to know how that would come out and the exact details and structure of such a law would matter. A law limiting the extraterritorial jurisdiction of U.S. states trying to regulate abortion would probably be upheld under the commerce clause. On the other hand the constitutionality of a law simply legalizing abortion in every U.S. state would be harder to determine.