He warned that if the Government goes "any further, the entire scheme will collapse."
Sunak claimed the legislation "blocks every single reason that has ever been used to prevent flights to Rwanda from taking off".
But he added: "The only extremely narrow exception will be that if you can prove with credible and compelling evidence that you specifically have a real and imminent risk of serious and irreversible harm.
"We have to recognise that as a matter of law, and if we didn't, we'd undermine the treaty we've just signed with Rwanda.
WATCH: Braverman warns the Tory Party is facing 'electoral oblivion'
"As the Rwandans themselves have made clear, if we go any further, the entire scheme will collapse.
"And there is no point having a bill with nowhere to send people to.
"But I'm telling you now we have set the bar so high, that it will be vanishingly rare for anyone to meet it."
He addressed the public from Downing Street in an attempt to defend his new Rwanda legislation.
In his resignation statement, Jenrick said he "cannot continue" in his post given he has "such strong disagreements with the direction of the Government's policy on immigration".
He continued: "The Government has a responsibility to place our vital national interests above highly contested interpretations of international law.
"In our discussions on the proposed emergency legislation you have moved towards my position, for which I am grateful. Nevertheless, I am unable to take the currently proposed legislation through the Commons as I do not believe it provides us with the best possible chance of success. A Bill of the kind you are proposing is a triumph of hope over experience.
"The stakes for the country are too high for us not to pursue the stronger protections required to end the merry-go-round of legal challenges which risk paralysing the scheme and negating its intended deterrent."
Jenrick warned: "I refuse to be yet another politician who makes promises on immigration to the British public but does not keep them."
His resignation came just minutes after the Government published legislation to address the concerns of the Supreme Court when they ruled that the Rwanda Plan is illegal.
But a source close to former Home Secretary Suella Braverman warned that the legislation will allow "every single illegal migrant to make individual human rights claims", saying it is a "further betrayal of Tory voters".
The source said: "This bill doesn’t come close to meeting Suella’s tests.
"The PM has kept the ability for every single illegal migrant to make individual human rights claims against their removal and to then appeal those claims if they don’t succeed. It is fatally flawed.
"It will be bogged down in the courts for months…. It won’t stop the boats.
"It is a further betrayal of Tory voters and the decent patriotic majority who want to see this insanity brought to an end."
Sunak's press conference also saw him say that next week's vote on his Rwanda legislation would not be treated as a confidence matter.
Asked whether he would remove the whip from Conservative MPs who voted against the Bill, he said: “No, but what this vote is about is about confidence in Parliament to demonstrate that it gets the British people’s frustration.
“I get it, I’m acting on it. So actually the real question when it comes to all these votes is for the Labour Party, because I want to get this legislation on the statute books as quickly as possible. That’s what we’re all about.
“We’ve moved at record pace since the judgment to get the treaty, to get the Bill introduced, so the question now is for the Labour Party.”
He added: “This is our deterrent, we are doing everything we can to get it on the statute books and get it up and running, so the question for votes in Parliament is what is the Labour Party’s plan and are they going to back this legislation?”