The CAA had agreed to close the skies over the hotel in northern Israel.
But Transport Minister Yisrael Katz then threatened to sack the CAA boss if he did not overturn that decision.
In the end, the skies stayed open to all air traffic during Thursday's wedding, Israeli media reported.
The 30-year-old Refaeli, Israel's most famous supermodel, married billionaire Israeli businessman Adi Ezra at the Carmel Forest resort near Haifa.
According to Israeli media reports, the couple was planning to have five drones, two helicopters and an observation balloon circling the venue, some for photographic purposes.
The CAA initially agreed to the couple's request for a ban, citing safety reasons.
But Mr Katz on Sunday overturned that decision saying: "The skies belong to all Israeli citizens and we can't offer special treatment for this event over other events."
Israeli pilots had also objected to the ban.
Two days later, according to Israeli media, the CAA, appearing to disregard Mr Katz's comments, announced a no-fly zone would be in place for safety reasons from 17:00 on Thursday to 02:00 on Friday.
Mr Katz reportedly told CAA chief Joel Feldschuh: "If the skies are not open above Bar Refaeli's wedding - you will be removed from your post."
On Thursday evening the transport ministry told The Jerusalem Post that the skies had stayed open to all air traffic.
The Times of Israel reported the rabbi was flown in by helicopter for the ceremony.
Forbes Israel has ranked Ms Refaeli Israel's top-earning model reporting revenue of nearly Â£10m in 2013 when she appeared in a lucrative Super Bowl advert.
She has hosted Israeli X Factor and modelled for brands including Desigual and Escada.
A former girlfriend of actor Leonardo DiCaprio, she prompted criticism in 2003 when it was reported she had opted not to finish her compulsory military service.
In 2010, a no-fly zone was put in place for the wedding of Chelsea, daughter of former US President Bill Clinton.
A request by Israeli supermodel Bar Refaeli to close the air space over her wedding venue has sparked a row between the country's transport minister and the civil aviation authority (CAA).