Upvote:1
Although the USA Today is not a government institution, it seems the official page corroborates the information: you don't need a visa but you do need a valid passport.
Green card holders visiting the British Virgin Islands are granted entry without visas as long as travelers meet the standard travel requirements. Visitors are allowed into the islands for a month provided the correct documents are shown at a land or sea port of entry: a passport, proof of accommodations on the island such as a hotel reservation and proof of your return journey such as an airline ticket. If you're a green card holder, you'll need to use a passport from your country of origin.
Emphasis mine.
BVI Tourism somewhat corroborates although doesn't mention a green card:
European citizens need a valid passport to enter the British Virgin Islands.