The Transport Secretary added that the millions who voted to Leave the European Union would feel "cheated" and putting a stop to Britain's withdrawal from the EU could end centuries of "moderate" politics the United Kingdom has enjoyed since the English Civil War.
Theresa May faces nearly certain defeat when MPs vote on the deal on Tuesday.
Labour Party leader Jeremy Corbyn is planning to call for an immediate vote of non-confidence in Theresa May's government on Tuesday evening assuming her Brexit withdrawal bill is heavily defeated in Parliament.
"When you turned out to vote in the referendum, you did so because you wanted your voice to be heard". We can not - and must not - let you down.
"Doing so would be a catastrophic and unforgivable breach of trust in our democracy", May said. "My message to Parliament this weekend is simple: It is time to forget the games and do what is right for our country".
Mrs May's warning came as Downing Street was reported to be concerned about efforts by a cross-party group of MPs to change Commons rules to enable backbench motions to take precedence over government business if Mrs May's deal falls.
She said the United Kingdom risks crashing out of the European Union without a deal or, if MPs are "unwilling" to face the uncertainty of no deal, then the United Kingdom may not leave at all.
It was previously thought only ministers could extend the two-year Article 50 process which, which governs how a member state leaves the EU.
The Cabinet ministers were speaking as MPs prepared for the third day of debate in the House of Commons ahead of next Tuesday's crunch vote, with Home Secretary Sajid Javid opening proceedings which are expected to be dominated by the issue of migration.
Britain and the European Union reached a hard-won Brexit deal in November, but the agreement has run aground in Parliament.
Labour peer Lord Andrew Adonis, who is a prominent Remainer, slammed Grayling and in a tweet wrote: "At the end of Brexit there should a one clause bill banning Grayling from holding any further public office ... unless of course it's overturned by a people's vote". This week, lawmakers passed an amendment forcing the government to come back to Parliament with a new plan within three working days of the deal being rejected. However, with more than 100 Tory MPs opposing the deal, it is expected to be rejected. "Projectfear? Project terror? Extremism more likely if we get failed Brexit which leads to lower living standards and less money for services".