The aircraft was returning from Lanzarote when it encountered difficulties trying to touch down during strong winds on Tuesday afternoon.
The pilot attempted to land the flight, code FR2047, before aborting with the aircraft just metres from the ground.
He climbed again when around 20ft from the runway and attempted a second landing, but this too was terminated and the plane instead diverted to Manchester Airport.
Passenger Amanda Hizzett, who was travelling with six members of her extended family from the Doncaster and Normanton areas, said that fellow passengers were 'frightened' during the drama.
"The flight was nearly full. We left Arrecife at 7.10am and the captain said it would be approximately three hours 20 to three hours 40. When we came in to land it was very bumpy and the plane was moving from side to side.
"We got to about 20ft from the runway and suddenly started to climb again as we were coming in at an angle to the left. We went back up and the pilot circled and started to come down again - however the second time he didn't get as far down and took off again and told us we had to divert to Manchester. There were people being sick and they were quite frightened."
The aircraft waited on the tarmac at Manchester for around an hour before passengers were allowed to disembark. They were taken by bus back to Leeds Bradford.
"We arrived home five hours late, but relieved to be in one piece thanks to the pilot's skills and the difficult decision he made to abort the landing," added Amanda.
Amanda's son Aron filmed the attempted landing from his seat in the cabin.
Leeds Bradford Airport confirmed there were no other diversions as a result of the weather conditions.
A statement from Ryanair read:
“This flight from Lanzarote to Leeds Bradford (17 Oct) performed a routine go-around and diverted to Manchester due to extreme winds at Leeds. The aircraft landed normally and customers were transferred to Leeds by coach. Ryanair sincerely apologised to all customers affected by this weather diversion, which was entirely beyond our control.”