After 39 minutes of deliberation, the tribunal consisting of chairman Jeff Gleeson, Shane Wakelin and Jordan Bannister, upheld the ban, ruling Archer out of games against Melbourne, Adelaide and Sydney.
Later on Tuesday, Richmond will challenge Tom Lynch's one-match suspension for rough conduct on Carlton's Tom De Koning.
Then, Hawthorn will challenge defender Jack Scrimshaw's three-match ban for a high hit on Essendon's Jordan Ridley.
Dogs defender Cleary was knocked out when Archer's knee made contact with his head in a contest in Saturday night's match at Marvel Stadium.
Play was stopped and Cleary received medical attention before being taken to hospital.
The match review officer graded Archer's actions as careless conduct, severe impact and high contact, resulting in a three-game suspension.
Cleary, 23, was back at Whitten Oval in good spirits on Monday but will miss the Bulldogs' match against Collingwood on Friday night under concussion protocols.
North's defence centred around Cleary's decision to go to ground to collect the ball, rather than stay on his feet.
Gleeson acknowledged rules encouraged players to keep their feet but this didn't always happen and "players need to be aware", saying Archer had "slowed too little and too late".
North can still take the case to the AFL appeals board.
Archer said he was initially competing for the ball, then looked to press.
"I expected him to pick up the ball and stay on his feet," he said, noting players were trained not to go to ground.
Archer said once Cleary's knee hit the ground, he had slowed down and braced for impact.
"My bum's down, trying to slow down as much (as possible), hit the brakes," he said.
Archer reached out to Cleary the following day and the Bulldogs defender had told him the incident wasn't his fault.
AFL lawyer Andrew Woods said Archer hadn't taken "reasonable care" to avoid the collision and had Cleary stayed on his feet and bent down to get the ball, the outcome could have been worse.
North used graphs showing Archer's "active deceleration" prior to impact and lawyer Justin Graham noted he showed "no indication of ever turning to bump" or "leaving the ground".
Graham said Cleary also appeared to receive contact in his back/shoulder from Jacob Konstanty, who was behind him.
Melbourne are yet to decide whether they will accept or challenge Aidan Johnson's one-match suspension for rough conduct on Giants forward Callum Brown.
Sydney have accepted Justin McInerney's three-match ban for the bump that concussed Brisbane's Brandon Starcevich.
