Clayton Bottomley was rushed to hospital after a mast climber is understood to have failed dropping the victims several floors at the Unity Street site.
His father David,53, was confirmed dead at the scene.
Both men were from Castleford in West Yorkshire.
Merseyside Police said: “Following an incident at a building site on Chapel Street on Wednesday 19 May, we can confirm that a 17-year-old male sadly died in hospital on Sunday 23 May.
“The male’s next of kin have been informed and are being supported by specially trained officers at this time.
“A second man, aged 53, sadly died at the scene of the accident.
“Emergency services were called to a site around 3.30pm to reports that the two men had been found seriously injured after falling.
“An investigation is being carried out by detectives and the Health and Safety Executive.”
The Yorkshire Post reported a statement from a member of Lock Lane Rugby where Clayton was a junior player.
It said: “I write this with a heavy heart to inform you all that one of our junior players Clayton Bottomley has sadly passed away after a horrific accident.
“Clayton joined Lock Lane in the Under 12 and played through to the under 16’s
“He was raw at first but soon got into the game, a very determined player to do well. He never took a backward step and gave 100% every game. He used to bring his large speaker on match day to get the team morale up in the changing room, he was a true team player.
“No parent should have to lose their child and our thoughts and prayers go out to his mum Paula and his sister Molly who he adored.
“This has had a massive impact on our team, our club and the whole community.”