Rudd was a moderate member of former prime minister Theresa May's government whose endorsement Johnson coveted during his successful United Kingdom leadership challenge in July.
Parliament will be suspended for five weeks and will not return until October 14.
"The government is expending a lot of energy to prepare for "No Deal" but I have not seen the same level of intensity go into our talks with the European Union", she wrote in her letter to Johnson, adding that she "no longer believes leaving with a deal is the government's main objective".
The departure of Ms Rudd - who complained she had seen little evidence Ministers were trying to find an agreement with Brussels on a new Brexit deal - prompted speculation that other senior Ministers might quit.
"It is unchanged, we will be leaving on October 31", he said.
MPs will vote on a motion calling for a snap election later, but it is not expected to get the two-thirds majority required to pass.
Such a high profile resignation from his heart of his cabinet will come as a huge blow to Mr Johnson, who has been fighting a rearguard action since MPs took control of the Commons this week to force him to seek an extension to Brexit on October 31.
Here is a breakdown of what has happened this week in the of commons.
The former Ukip leader is demanding that Brexit Party candidates are allowed to run against Labour candidates without Conservatives to contend with in the North, the Midlands and South Wales.
The pact aims to secure the UK's exit from the European Union, as well as place Farage as the leader of a Westminster party with a considerable number of seats.
Asked about the possibility of an electoral alliance with the former UKIP leader, Chancellor Sajid Javid did not categorically rule out the idea, but said the Conservatives "don't need an electoral alliance with anyone".
Over the summer, Boris Johnson said he accepted the onus was on the United Kingdom to come up with alternatives to the existing plan.
Julian Smith today moved swiftly to dismiss suggestions he would be the next Cabinet minister to quit in protest at a brutal purge of Tory Remainers.
"Anyone who understands how negotiation works, [knows] you would not discuss those in public", he added.