More than 1.3 million signatures have been added to a petition calling for Parliament not to be prorogued, amid growing anger over the decision.
Thousands of people rallied for hours outside Parliament on Wednesday night, and there were smaller demonstrations in other towns and cities.
The plan to suspend Parliament was heavily criticised by opposition parties, and has been criticised by some Conservative MPs, and there are legal challenges in English and Scottish courts.
The e-petition on Parliament’s website gained the 100,000 signatures needed for it to be debated by MPs in just a few hours, and reached a million before midnight.
On Wednesday, Mr Johnson said he wanted to prorogue Parliament in order to bring the current record-breaking session to a close in order to bring forward his Government’s new legislative agenda.
But opposition leaders said the Prime Minister is trying to halt their efforts to block a no-deal Brexit.
The petitioners say Parliament “must not be prorogued or dissolved unless and until the Article 50 period has been sufficiently extended or the UK’s intention to withdraw from the EU has been cancelled”.
Legal challenges against Mr Johnson’s decision are mounting, with separate bids launched in London and Edinburgh courts seeking an emergency injunction to prevent Parliament being suspended.
It comes as impromptu protests sprung up across the UK in London, Edinburgh, Cardiff, Manchester, Bristol, Cambridge and Durham against the move.
ITV News Scotland Correspondent Peter Smith said sources had told him Ms Davidson's decision was not due to the Government's decision to suspend Parliament, but was something she had been considering for a while for “a mix of personal and political reasons, with emphasis on the personal".
He added that the MSP's political and personal differences to the Prime Minister may have played a part.
Barrister Tom Hickman tweeted that anti-Brexit campaigner Gina Miller “has issued proceedings to challenge prorogation. Mishcon de Reya, Lord Pannick QC, Warren Fitt and me acting”.
Speaking to BBC News, Ms Miller said the PM was “hijacking the Queen’s prerogative power” and using it for “unscrupulous means”.
She added: “I think that is what so shocking about this, is that its a very cowardly way of using these powers and constitutional convention.
“Our unwritten constitution is a bit like a gentleman’s agreement, and you have to say it’s not been used in that manner.”
In 2016, Ms Miller launched a successful legal bid, with judges ruling that MPs would have to vote before the Government could invoke Article 50 to formally start the UK’s exit process from the EU.
A separate bid has been launched by pro-Remain barrister Jo Maugham, director of the Good Law Project, who has filed a motion asking the Scottish Court of Session to suspend the PM request that Parliament be prorogued.
Former Tory prime minister Sir John Major also said on Wednesday he is seeking advice on the legality of Mr Johnson proroguing Parliament.
ITV News Political Correspondent Daniel Hewitt explains what happens next after Mr Johnson has decided to prorogue Parliament
MPs will return to Parliament on Tuesday, but just over a week later on September 10, at the earliest, Parliament could be prorogued until October 14 ahead of a Queen’s Speech.
Mr Johnson’s prorogation plan came just a day after opposition leaders struck a deal to try to block a no-deal Brexit through legislative means.
Labour leader Jeremy Corbyn called it an attempt to "ride roughshod" over Parliament and prevent any legislation or debate that would stop a no-deal Brexit.
He said: "He seems to want to run headlong into the arms of Donald Trump with more determination than I've ever seen in anyone else before.
"This is extraordinary. He needs to be held to account by Parliament, not by shutting down Parliament, but by attending Parliament and answering the questions."
Liberal Democrat leader Jo Swinson told the BBC’s Newsnight programme that anti-no-deal MPs could use “arcane and unusual” legislative routes to try to block the UK leaving the EU without a deal.
She added: “There are legislative avenues being explored between cross-party groups of members of Parliament with legal and constitutional experts looking at how this can be done, and different routes.
“We have got a Government that is prepared to take unprecedented routes and so we are looking at options as well that might be arcane or unusual that could be employed.”
Eddisbury Tory MP Antoinette Sandbach told BBC Newsnight: “I think it’s deeply worrying and to my mind very, very concerning that we are having such a long prorogation, that is what concerns me.
“My constituents want to have answers to their questions and this, the length of time, stops it. There is nothing to stop Boris Johnson from bringing forward domestic legislation in the normal way and it would have been possible to prorogue just for a short period, three days maybe.”
The Welsh Assembly will be recalled early to hold a day of debate on September 5 to discuss what First Minister Mark Drakeford called the “constitutional crisis which now faces Wales and the United Kingdom”.
The Scottish Parliament will sit on Monday at the end of its summer recess, with a debate on Brexit expected during the first sitting week.
Ms Davidson is expected to make a statement on Thursday where she will announce that she is quitting as leader of the Scottish Conservatives.
The former journalist, who took over the top job in the party almost eight years ago, will “make her position clear in due course”, a spokesperson said.
But it is expected she will step down on Thursday, with her resignation coming in the aftermath of Mr Johnson announcing plans to temporarily suspend Parliament.
Opposition leader Jeremy Corbyn is due in Scotland to start a three-day tour on Thursday.
The UK Labour leader will begin the campaign visits to Scottish constituencies by visiting a community centre in Dunfermline.