Police have not stated if they believed the two incidents were linked but have asked anyone with information to contact them.
The alleged author of the inscriptions was arrested on the public highway by the local police.
The swastika was covered up with tar a few days after its appearance was reported.
This vandalism and attack in Abu Ghosh have appeared amid a rapid increase in settler violence since a Palestinian terror attack where four Israelis were killed outside of a gas station in Eli.
Italy's Carabinieri police said they had been helped to identify the couple by photographic evidence but did not name them.
He can be seen smiling, even as the author of the video, speaking in English and using swear words, reprimands him.
The carving features three distinct elements - a human face, a cornucopia and a phallus - within a space measuring 57 x 42 centimeters.
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