You don't need a Green Card to travel in the EU, plus some other countries, but you must be insured.
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All US green card holders should note that having permanent residency in the US doesn't qualify them for visa-free entry into Spain; only US citizens with US passports can enjoy such benefits. However, they can apply for a visa by visiting the Spanish embassy in their home country.
You can travel without a green card because you are already a lawful permanent resident. Upon arrival at the port of entry, U.S. Customs and Border Protection (CBP) officers will stamp your passport with an I-551 stamp. The I-551 stamp is also called the Temporary Proof of Legal Residence stamp.
In much of Europe, the only travel document a US or Canadian citizen needs is a passport. (The US Passport Card works only for those driving or cruising to Canada, Mexico, Bermuda, and the Caribbean.)
Americans Will Need a 'Visa' to Visit Europe in 2024—Here's What to Know. The new program requires an online application and a minor fee, and is required for all travelers regardless of their age.