To my ear, the tune is not sol-mi-7-sol-mi. The 7 is too high. This accords with the fact that the harmonic seventh is more than twice as far from the equal tempered major sixth (a difference of 68.83 cents) as it is from the equal-tempered minor seventh (a difference of 31.17 cents).
One could make a case for it being 7-sol-do-7-sol, however. This makes the first interval, the descending minor third, a 7:6 ratio (or 6:7 if you prefer, since it's descending). That's 266.87 cents, while a 6:5 minor third is 315.64 cents, and an equal-tempered minor third is of course 300 cents. It also makes the interval between the higher two pitches a very large 8:7 step (231.17 cents) instead of a 9:8 step (203.91 cents) or 10:9 (182.40 cents).
Tuning the A of Salve regina to the harmonic seventh also sounds horribly wrong to me. If it was ever sung that way, it would have had to have been long before the early 11th century, when Guido d'Arezzo codified the basis of hexachord-based solmization. But according to Wikipedia, it seems to have been composed in the 11th century, but the "solemn tone" given there is completely different from the tune you link to, which I suppose is even more recent than that. Perhaps you could find evidence for this hypothesis in an older chant.
It does seem fairly well established that medieval keyboards (organs, for example) were tuned in untempered fifths, though I don't know as much about this as I know about baroque keyboard temperaments. Untempered fifths work if you're not playing music with consonant thirds, and your keyboard only has one or two "black" notes (B♭ and F♯). This puts the A at a ratio of 27:16 over C, or 905.87 cents. That's a lot closer to equal tempered A at 900 cents than it is to B♭ at 1000 cents.
Furthermore, the harmonic seventh is closer to C than it is to G, but the note is written using the lower of the two diatonic steps between those two notes. If the harmonic seventh had really been in use, it would seem much more likely that the note would have been written as a B♭ than an A.