Irsay is the first team owner to publicly support the possible removal of Snyder, who is facing sexual misconduct allegations.
A bombshell report from ESPN last week said Snyder believed he had enough "dirt" to "blow up" some of his fellow owners and NFL Commissioner Roger Goodell, as he faces renewed calls to sell his team.
The NFL fined the Commanders $10 million last year after an independent investigation showed widespread bullying and sexual harassment of female employees at the team.
In February, the league said it would independently investigate after a former cheerleader and marketing manager for the team accused Snyder of inappropriate behavior.
Snyder has denied the allegations.
"It's a regrettable situation," Irsay told reporters. "It pains me to see it - the founders of this league taught me, 'You have to protect the game, you have to protect what we're about.' This isn't what we're about."
A spokesperson for the Commanders said that Snyder would not sell the team.
"It is highly inappropriate, but not surprising, that Mr. Irsay opted to make statements publicly based on falsehoods in the media," the spokesperson said in a statement.
"We are confident that, when he has an opportunity to see the actual evidence in this case, Mr. Irsay will conclude that there is no reason for the Snyders to consider selling the franchise."
In a letter to team owners obtained by Reuters, Snyder hit back at the ESPN report, saying that he had never hired investigators to "look into other owners" and Goodell.
"That is patently false and intended to erode the trust and goodwill between owners," he wrote.
Team owners did not vote on the matter on Tuesday but Irsay said he believed there was enough support for Snyder's potential removal, pending the outcome of the NFL investigation. It would take the support of at least 24 of the league's 32 owners to oust him.
"I'm very concerned that he needs to be removed," said Irsay.
"It's a situation that you wish wasn't there."
Goodell told owners there was "no reason for there to be any speculation" before the inquiry, led by former Securities and Exchange Commission chair Mary Jo White, was complete.
"Speculation without facts is not a very positive thing to do. I think everyone deserves to have facts and make sure those decisions are made with facts," he told reporters.
Goodell said he was unable to provide a timeline on when the investigation would be complete but reiterated that the findings would be made available to the public.
