There is no European tourist visa. Each country you want to visit issues the visa for its own country, the visa issued by France is also valid to visit 23 other countries on the continent but not in the United Kingdom. It depends on your itinerary which visa you need to apply for at first. If you plan to travel U.S. - London - Paris - U.S. you should apply for the French visa at first showing a flight reservation back to the United States as your intend to leave the country. After it has been approved you can apply for the United Kingdom visa showing a flight or train reservation to France as proof of leaving the country. If you travel the other way around you should first apply for the U.K. visa. If you book a round trip U.S. - U.K. and you plan a side trip to France you should first apply for a U.K. double entry visa showing a return flight reservation back to the States and afterwards apply for a single entry visa to France showing a flight or train return ticket back to London.

The U.K. visa application form must be filled and paid online, afterwards you will need to make an appointment with USCIS to have your biometrics and fingerprints taken. THen you can mail your printed out application together with the supporting documents to the U.K. Border Agency at the Consulate General in New York which processes all visa application made from northern/central America and the Caribbean. You can apply for the visa up to 3 months prior to the proposed travel date.
https://www.visainfoservices.com/Pages/dest_org.aspx ..

The visa for France must be applied for in person at the French Consulate General which covers your U.S. state of residence. It can be applied for up to 90 days prior to travel.
http://www.ambafrance-us.org/spip.php?rubrique=2 ..

The U.S. Green Card does not allow you to travel without visa, to do so you must get naturalized and get a U.S. passport once you qualify. It just enables you the re-entry to the U.S. however it eases up the risk assessment of illegal immigration. Lawful permanent residents in the United States or Canada normally present a very limited risk of illegal immigration to Europe.