Upvote:0
Per https://www.uscis.gov/green-card/after-green-card-granted/international-travel-permanent-resident:
If seeking to enter the United States after temporary travel abroad, you will need to present a valid, unexpired βgreen cardβ (Form I-551, Permanent Resident Card). When arriving at a port of entry, a U.S. Customs and Border Protection Officer will review your permanent resident card and any other identity documents you present, such as a passport, foreign national I.D. card or U.S. Driverβs License, and determine if you can enter the United States.
So short answer: you don't need the UK passport specifically to help establish that you are the person for whom the green card was issued, but the UK passport is an option.
Somewhat related: the US does frown on dual citizens using a foreign passport to enter or leave the US (https://travel.state.gov/content/travel/en/legal-considerations/us-citizenship-laws-policies/citizenship-and-dual-nationality/dual-nationality.html):
Most U.S. nationals, including dual nationals, must use a U.S. passport to enter and leave the United States. Dual nationals may also be required by the foreign country to use its passport to enter and leave that country.
Summary: the green card and some photo ID is sufficient. In light of the US's attitude to its citizens (which of course are distinct from permanent residents) entering on foreign passports, I'd probably consider showing the UK passport to be the last resort.
Upvote:2
You don't need a passport to reenter the US, for trips of less than 6 months you only need your green card. The passport may be required by by the countries you travel to, so only their requirements matter.