Actually, even if there were substantial numbers of spanish immigrants, I'd say they would identify themselves as galician/asturian/catalan/whatever first and spanish later; and spanish identity would end diluting between their "original" regional identity and their adopted american identity. Something similar happened in Cuba, where there are huge galician and asturian communities that have never really identified themselves as "spanish".
IIRC, there is a sizeable galician community in New Jersey today in the Newark area.
And don't forget basques in Boise.
I agree mostly with Dr. Strangelove on this, though I think that also some of them could indentify themselves among other hispanics. In OTL, at least some famous americans with spanish ascendancy did so. Also like others said, spanish will prefer Argentina, Cuba and Venezuela (in this order as in OTL, if nothing impedes it). Even nowadays latin american emigrants prefer Spain to USA due to cultural reasons despite economical oportunities, excepting, of course, those who live next door to the USA like Mexicans and central americans.
If for some reason, spaniards can't go to those countries, you have still Uruguay, Paraguay, Mexico, Perú, Colombia, Brazil etc, which also saw spanish inmigration in OTL. So, you need to close the entire Latin America for spaniards in order to force them to emigrate to USA.
PD: Even in Madrid and its periphery, where tons of inmigrants from other parts of Spain came (like my parents), inmigrants' institutions are strongly regionalized, and you have separate cultural houses for galicians, andalusinas, asturians, extremenians etc.