Dual Passport Situation

I have a US and a EU passport

You MUST (certainly by law in the USA) show the passport pertaining to teh country you enter or leave.
In our case, we must show the US passport at the US border both ways and in Spain the same. When traveling to the USA you also be asked to show your US passport and often they have it on record in their computers.

Since you do not need a visa with both those passports to be shown, there is no matter of getting any exit/entry stamp in play.

As to losing your nationality, US nationality cannot be lost (biggest pain in teh a***, worse US citizens are almost the only ones who owe teh US government eterally so all their lives taxes regardless of where they live, a very rare tax regime that is constitutionally illegal since laws are territorial in nature not global but enforced anyway).

In Europe several countries have laws specific only to their country about loss of nationality. Dutch nationsl (for example) lose their nationality after living 10 years outside the EU. So your risk is biggets with your Spanish passport. Make sure to check if you are still Spanish before going there. And no, as passport is NOT the same as "having a nationality". Good legal advice is important anyway since the consequences of losing one's nationality are often dire.

This may get you started on this subject as far as your Spanish nationality goes.

https://www.mjusticia.gob.es/cs/Satelli … cionalidad

Dennis, whilst there are a few exceptions, most people who obtain a Spanish passport are required to renounce all other nationalities.  In
those cases it is a criminal offence to have a spanish PP and another.

NB     I did say there are a few exceptions.  You may be one of those, if so please ignore this post

Regards.  John