There are no activities I can find that take place on the Israeli side of the Green Line.
For example, the Consulate in Jerusalem offers 25 Fulbright Scholarships - with the following condition: applicants must be "legal residents of Gaza Strip, West Bank and East Jerusalem."
Here is a list of events:
# Burns and Hughes Co-host a Ramadan Iftaar ( )
# Cinderella in Jericho ( )
# Consulate General host a special Iftar Dinner in Ramallah ( )
# Iftar and Storytelling for Shoufat Camp Children ( )
# U.S Consulate Jerusalem Hosts Ramadan Quiz Nights ( )
# YES Alumni Address ACCESS and High School Students in Ramallah (October 5, 2006)
# Ramadan Night at the American Corner in Jericho ( )
# U.S. Consulate Hosts Religious Tolerance Dialogue between Prominent Palestinian and American Muslims (October 4, 2006)
# Ramadan Storytelling and Music for Children ( )
# Concert by Al Kamandjati at the Palestinian National Theater ( )
# Celebrating Ramadan ( )
# USAID Provides Scholarships to Over 2000 Palestinian Students ( )
# U.S. Consulate Reinvigorates Partnership with the Jericho Municipal Library (August 17, 2006)
# Consulate Brings Musician/Environmentalist TH Culhane to Power up Summer Camps in East Jerusalem (July 19, 2006)
# U.S. grant creates Opportunity for Palestinian Women to promote their handicraft skills ( )
# Over 700 Palestinian Youths graduate from U.S.-funded English language training program ( )
# Gazan Students Graduate from English Language Program (June 5, 2006)
The site is available in English and Arabic - but not Hebrew. (Even the US Embassy site in Tel Aviv doesn't provide a Hebrew page.)
The Consulate is physically located in West Jerusalem - on the Israeli side of the Green Line.
The United States Congress agreed to move its Israel embassy to Jerusalem in 1995 by an overwhelming vote, but every president since then has stopped the move. Yet it appears that the US State Department does recognize Jerusalem as the capital of a nation, even if it doesn't have an official embassy there.