Vatican City is indeed an independent State; but it is fully surrounded by the city of Rome completely within the State of Italy. As a result, Roman and Italian authorities have a great deal of control of the facts on the ground. Having an independent City-State of Jerusalem on the fault-line between two countries who have been bitterly disputing where that boarder should be for year, seems like a recipe for disaster, especially considering the history of armed militant factions that currently operate within Palestinian areas.
UN aid chief says no warring parties observed Yemen truce (Reuters)
from the article: Would East Jerusalem Arabs rather be citizens of Israel or Palestine?