The victim is in a London hospital after being found on a fifth-floor roof, the Metropolitan Police said.
A police spokesman told PA that the child is "no longer in a life-threatening situation".
"He is critical, but stable," the spokesman added.
The 17-year-old male suspect had remained with members of the public on the 10th-floor viewing platform after the incident on Sunday afternoon, the force said.
They added that there was nothing to suggest that he is known to the victim.
Officers had been called to the gallery at around 2.40pm and the child was treated at the scene before being flown to hospital by London's Air Ambulance.
Visitors reported on social media not being allowed in or out of the gallery while emergency services dealt with the incident.
Scotland Yard said a number of members of the public are assisting police with witness statements.
Administration worker Nancy Barnfield, 47, of Rochdale, was at the 10th-floor viewing gallery with a friend and their children when her friend heard a "loud bang".
Ms Barnfield turned around and saw a woman screaming "where's my son, where's my son?"
Members of the public quickly gathered around a man who was nearby, she said.
Ms Barnfield said: "We did not notice the mum before, we noticed her after because she was hysterical by then."
She said the person who was restrained by members of the public before the police arrived "just stood there and was quite calm".
But most visitors only discovered that someone had been injured as they tried to leave.
A visitor, who did not want to give his name, said they could hear an air ambulance which had landed on the concrete walkway in front of the building.
A group of uniformed police officers could be seen going into one of the gallery entrances at 4.53pm as the metal shutters were brought down and visitors were turned away.
BBC journalist Jonny Dymond, who was in the gallery at the time of the incident, said visitors were funnelled into a main hall while all exits were closed.
He added: "There were quite a lot of families with children, and security guards told us we couldn't leave.
"There were at least two fire engines, 10 police cars and an incident control unit.
"Parts of the exterior of the building were taped off."
Tate Modern remained closed for the rest of Sunday but will open on Monday as usual although the viewing platform will remain closed, a spokeswoman for the Tate said.
A photocall for a new exhibition which had been due to take place on Monday morning has been postponed, she added.
The spokeswoman said: "Tate is working closely with the police to help with their investigations.
"All our thoughts are with the child and his family."
The Tate Modern was the UK's most popular tourist attraction in 2018 after being visited 5.9 million times, according to the Association of Leading Visitor Attractions.
With the schools having broken up and holiday season begun, there are likely to have been thousands of visitors to the gallery on Sunday.