What to do to maintain my Spanish ULM (GA) gyroplane licence valid in the USA when moving there?
As a caveat here, I know nothing about gyroplanes. But based on the US regulations it looks like there are a few different possibilities.

If your ULM license is equivalent to an ICAO private license then you could get an FAA foreign-based private license as described in this question. I believe you would get category rotorcraft and class gyroplane (see 14 CFR 61.5) but I'm not completely sure.

If your ULM license is not equivalent to an ICAO private license then it depends on the type of gyroplane you want to fly. If it's considered an ultralight then you don't need any license at all. If it's an LSA then you would need a sport or recreational pilot license. If it's a 'regular' gyroplane, you would need a private license.

If you do need to get an FAA license (sport, recreational or private) you should be able to use time you logged in Spain to count towards it.