passport validity rule in Spain, Greece & other European. counties

Your arrival in Barcelona is the only point of concern as that is the point of entry. It won't be a problem but given the amount of time between now and when you travel, why aren't you getting them updated?

This is the official EU website: https://europa.eu/youreurope/citizens/travel/entry-exit/non-eu-nationals/index_en.htm

If you are a non-EU national wishing to visit or travel within the EU, you will need a passport:
- valid for at least 3 months after the date you intend to leave the EU country you are visiting,
- which was issued within the previous 10 years,

10th July + 3 months = 10th October. You are legal.
Since you can stay in the Schengen Area for 90 days, there have been reports of some airlines miss-interpretting this as 6 months from arrival date (they assume you will stay the full 90 days). Even if you get someone at check-in who has been mis-trained:
22nd June + 6 months = 22nd December. You are still OK.

Thanks for the fast reply. Glad to hear that our passports are still valid. We are American citizens. I’m just afraid that we won’t get our passport back in time if we renew it now. Does this countries needs visitors visa to enter?

You enter visa-free into these countries with your USA passport for the purpose of tourism (no work), for up to 90 days. As mentioned above, you'll only show the passport upon entry to Spain, not at other border crossings or airport arrivals, since all of them are coordinated under the Schengen Visa Zone. See also https://www.ricksteves.com/travel-tips/trip-planning/travel-documents.

You are OK and you have plenty of time to renew. Personally I would do it. We always renew at about the nine month mark. Just one less thing to be worried about it.