The sculpture is one of about two dozen similar sculptures from the Middle Ages that still feature on churches around Germany and elsewhere in Europe.
Patriarch Kirill is a close ally of President Vladimir Putin and has strongly backed the war in Ukraine.
Christians believe the cave, which used to serve as a Roman cistern, is where Jesus healed 10 lepers.
Amid recent tensions at the Gaza, Lebanon and Syria borders boiling over, Israel reportedly canceled travel permits for Gazan Christians for the upcoming feast.
The report described that history as one of "repeated dismissal or cover up of that abuse by the Catholic Church hierarchy."
Ukraine's government has accused the UOC of retaining ties with Moscow, and has opened criminal cases against over 60 of its clergy on suspicion of crimes including treason and collaboration
The UOC has been accused of maintaining links to the pro-invasion Russian Orthodox Church, which used to be its parent church but with which the UOC says it all broke ties in May 2022.
Video footage of the incident shared online disputes allegations that the attack on the church was committed by two "Jewish radicals."
Orthodoxy in Lithuania dates back to the 13th century, Bartholomew said, when it fell under the ultimate supervision of Constantinople.
The Church of San Roman, built in the year of 1062, is now fully visible for the first time since the 1960s.