There's no scientific study that shows that European genes survived.
Essentially all Native American mtDNA haplotypes are A, B, C and D.
There is a fifth mtDNA haplogroup, X, found among WESTERN Americans, but not EASTERN, which some speculated might be of European origin, but that was shot down when it was shown to be identical to X mtDNA in the Altai region of Siberia.
X didn't come from Spain and it didn't come from France. It came from people walking East from Central Asia across Berengia.
mtDNA isn't nuclear DNA, and doesn't tell much of anything (other than possibly suggesting arrival by sea, or arrival of different waves, or much greater antiquity of humans in the Americas, or colonization from east to west, rather than west to east), simply because it is amplified as it is passed to all offspring.
I saw a show on tv where a Japanese gal had her DNA identified and it came out 30% caucasion. Have you heard that the early Japanese settlers were Siberian and of European descent? I have always wondered this and wanted to verify it.