The site, in the city of Dimona in the Negev Desert, is part of the same complex housing the office of a local Rabbi.
The demonstration, captured in a video posted on Facebook, was followed by a crowd breaking windows and causing further damage at the homes of two congregation families.
The centre was opened by Albert and Esther Knoester, and their house was one of the two targeted, according to Kehila News Israel. Albert serves as an elder at the Nachalat Yeshua church in nearby Beersheba.
The protestors are believed to be associated with Yad L'Achim, a group which has previously targeted Messianic Jews and Christians across Israel with demonstrations and legal action.
Congregations of Messianic Jews in southern Israel have suffered violence at the hands of ultra-Orthodox Jews in the past.
More than 600 men and women stormed a baptismal services of the Nachalat Yeshua congregation in Beersheba in 2005, attacking church goers and damaging property.
According to the Christian organisation, Middle East Concern, local believers are appealing for prayer.