In the Wikipedia article Full Faith and Credit Clause in specific regard to marriage, it is said that:
Until the Supreme Court struck down all laws banning interracial marriage in 1967, a number of states banned interracial marriage and did not recognize marriage certificates issued in other states for interracial couples. The full faith and credit clause was never used to force a state to recognize a marriage it did not wish to recognize. However, the existence of a common-law marriage in a sister state (still available in nine states and the District of Columbia) has been recognized in divorce or dissolution of marriage cases.
This seems to imply that DC was treated as if it was a state, but FF&C was not used to restrict Virginia and other states with similar laws.
In regard to the public policy limitations of the FF&C clause, the same article, quoting Pacific Employers Insurance v. Industrial Accident says:
[T]here are some limitations upon the extent to which a state may be required by the full faith and credit clause to enforce even the judgment of another state in contravention of its own statutes or policy. See Wisconsin v. Pelican Insurance Co., 127 U.S. 265; Huntington v. Attrill, 146 U.S. 657; Finney v. Guy, 189 U.S. 335; see also Clarke v. Clarke, 178 U.S. 186; Olmsted v. Olmsted, 216 U.S. 386; Hood v. McGehee, 237 U.S. 611; cf. Gasquet v. Fenner, 247 U.S. 16. And in the case of statutes...the full faith and credit clause does not require one state to substitute for its own statute, applicable to persons and events within it, the conflicting statute of another state, even though that statute is of controlling force in the courts of the state of its enactment with respect to the same persons and events
I have not been able to find a reported case where the FF&C clause was used to give effect to a DC judgment or law, but I did find Rollins v. Rollins 602 A.2d 1121 (1992) where DC was required to extend FF&C to a Maryland judgment, although in this case the decision was that the judgment was not final and so did not apply.
Based on all this, I do not think the Loving case would have been substantially different had the marring occurred in another state, say New York, that permitted such marriages at the time. The Virginia statute at issue in the case certainly seemed aimed at such situations.