According to Haaretz, Americans for Peace Now has filed a brief with the United States Supreme Court, arguing against recognition of Jerusalem as part of Israel. They believe Americans born in the city should be listed as having been born in "Jerusalem", without the name of a country, rather than writing Israel as their place of birth. This is the case even if a person was born in the western portion of Jerusalem, the part under Israeli sovereignty since the War of Independence.
It is unclear whether the Israeli Peace Now agrees with the actions of its American sister organization. If it does, it's very troubling. An Israeli organization that not only does not recognize West Jerusalem as Israel's capital, but doesn't even recognize it as being a part of the State, in effect, does not recognize the results of the War of Independence. What is the difference between the the areas gained by Israel in 1948 which were intended for the Arab State in the Partition Plan, and the parts of the Corpus Seperatum of Jerusalem which Israel took over during the same war? Neither the Arab State or the Corpus Seperatum ever came into being, so there should be no difference.
It makes no sense not to recognize Israel's sovereignty over the western portion of Jerusalem while recognizing its sovereignty over Nazareth, Be'er-Sheva, Nahariya, Akko, Jaffa and Ramla - all cities intended to be part of the Arab state. They are now rightly recognized by all the countries of the world as legitimate parts of Israel - except for countries that don't recognize Israel at all.