He was injured at a friend's barbecue on 8 April and remains in hospital.
Jackson, 28, underwent neck surgery for what his region at the time described as a "non-rugby related injury."
"After hitting my head on the bottom I realised I couldn't swim to the surface because I'd lost movement in my legs and power in my arms," Jackson said.
Jackson, who has also played for Bath, Wasps, London Welsh and Doncaster Knights, said his father - a retired GP - and a friend realised immediately something was wrong.
They pulled him to the surface and stabilised him until the ambulance arrived to take him to Southmead Hospital in Bristol.
"After a number of MRI scans and X-Rays the Drs decided to operate at 2am to stabilise my neck as pressure was being put on my spinal cord," Jackson added in a Facebook post.
"In surgery they removed my shattered disc, relocated my vertebrae and fixed it in place with a metal plate.
"I woke up in ICU, luckily completely coherent, however no feeling below my neck other than limited movement in my right arm."
Bath-born Jackson has made 36 appearances for the Dragons since joining from Wasps in 2015 and signed a contract extension with the region in December.

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The Gunners were knocked out of both the FA Cup and Champions League this week.
Wenger's position has been questioned and his team's performances criticised but they won 2-0 at Goodison Park to stay third in the Premier League.
"I was pleased with how well they responded today," said Wenger.
"It was a very significant win for us and a mental test.
"I like this group, I like the mental attitude, it has been questioned and that hurts me," he added after Danny Welbeck and Alex Iwobi scored in the win over Everton.
"When you work with them every day, you see how much effort they put in."
Arsenal were top of the Premier League on 4 January and fighting for silverware in two cup competitions.
But their form has collapsed in recent weeks and Saturday's win was their first in four games in the Premier League.
They lost to Watford on Sunday in the FA Cup quarter-finals and a 3-1 defeat by Barcelona on Wednesday sealed their exit from Europe.
They are now 11 points behind leaders Leicester City and three behind second-placed Tottenham.
But Wenger does not think his side are out of the title race with eight games remaining.
"Why should it be a two-horse race with one team three points in front of us? Let's focus on performances and after that we will see how we finish," he added.
Newport Gwent Dragons forward Ed Jackson has revealed he suffered a serious spinal injury after diving into the shallow end of a swimming pool.
[NEXT_CONCEPT]
Arsene Wenger said it "hurt" that his players' attitude was questioned and believes Arsenal passed a "mental test" to beat Everton on Saturday.