Delivering a lecture in London, Bercow said on Thursday: "Not obeying the law must surely be a non-starter. Period," the BBC reported.
The law Bercow was referring to was passed on Monday before Parliament was prorogued for five weeks in the early hours of Tuesday. It forces the Prime Minister to seek a delay until January 31, 2020, unless a deal or no-deal exit is approved by MPs by October 19.
The UK is due to leave the European Union (EU) on October 31, but the new law passed by MPs could extend that deadline.
"The only form of Brexit which we will have, whenever that might be, will be a Brexit that the House of Commons has explicitly endorsed," he said.
"Surely, in 2019, in modern Britain, in a parliamentary democracy, we - parliamentar, legislators - cannot in all conscience be conducting a debate as to whether adherence to the law is or isn't required."
He called it "astonishing" that "anyone has even entertained the notion".
If the government comes close to disobeying the Act, the MP said that Parliament "would want to cut off such a possibility and do so forcefully".
"If that demands additional procedural creativity in order to come to pass, it is a racing certainty that this will happen, and that neither the limitations of the existing rule book nor the ticking of the clock will stop it doing so," he added.
Bercow has announced he will stand down as Commons Speaker and MP at the next election or on October 31, whichever comes first, the BBC reported.
Meanwhile, Johnson on Thursday insisted that the UK "will be ready" to leave the EU by the current deadline without an agreement "if we have to".
In response to the publication of the government's Yellowhammer document, an assessment of a reasonable worst-case scenario in the event of a no-deal Brexit, Johnson reiterated it was "the worst-case scenario".
"In reality we will certainly be ready for a no-deal Brexit if we have to do it and I stress again that's not where we intend to end up," he added.
( With inputs from IANS )