When Loving v. Virginia was decided in 1967, there were 16 states that still had anti-miscegenation statutes. Most of the these states capitulated immediately and repealed their statutes. However, several states did not, and judges from these states simply refused to recognize Loving.

So yes, the very same kind of arguments, rooted in theories of state sovereignty, were being used by local judges after Loving, just as they are today after Obergefell. Though it should be said that the interplay between the 10th and 14th amendments was far less settled in 1967 than it is in 2015, making those arguments at least slightly more tenable 45 years ago then they are today.

In any case, this open resistance continued for three years, but Nixon’s justice department finally took action in late 1970. They had been on the lookout for a good case from a public relations standpoint, and they found it when Army sergeant Louis Voyer and his fiancé Phyllis Bett were denied a marriage license. As described by the Court in United States v. Brittain:

On November 10, 1970, Sergeant Louis Voyer, a soldier stationed by the Army at Fort McClellan near Anniston, Alabama, and Phyllis Bett, a resident of the city, attempted to procure a marriage license from the office of Probate Judge G. Clyde Brittain. After filling out the application form and waiting several minutes, they were told by one of Judge Brittain's clerks that they could not get a license because it would be against state law. Miss Bett is a Negro, while Sgt. Voyer is a Caucasian.

In response, the federal judge in the Northern District of Alabama granted an injunction:

enjoining the State of Alabama, its officers, agents, employees, and their successors, and all those acting in concert or participation with them from enforcing or giving any effect to such laws.

The remaining states gave up quickly after this opinion was issued. As stated in Loving v. Virginia in a Post-Racial World: Rethinking Race, Sex, and Marriage, p. 138:

While the immediate reaction to Loving, for example, was not positive, it did not provoke sustained and serious efforts to overrule it. Nor did Loving engender the sustained resistance that the Brown decision spawned, perhaps because the choice of a marriage partner still devolved to the individual, with limited state regulation. Brown, on the other hand, required social engagement by nearly every family with a child in public schools.

Yet even when it became clear to the last holdout states in 1970 that enforcing their laws would impossible, it still took until 1999 and 2000 when South Carolina and Alabama actually repealed their laws. In Alabama’s election in 2000, over 40% of voters supported keeping the laws.

Finally, I would note that the problem has not entirely gone away. In 2009, a Louisiana Justice of the Peace refused to marry an interracial couple. Fortunately the public backlash caused the aging racist to retire his bench.