The Congress had issued a whip to all its 17 members to vote against the trust motion moved by Manohar Parrikar.
Sanjay Raut also said that despite the rout the Shiv Sena faced in Goa, its work would continue in the state.
Mridula Sinha has drawn flak from the Congress for inviting the BJP to form the government in Goa, despite the saffron party winning only 13 of the 40 seats against the Congress' 17.
The Congress said it would move a substantive motion and come back for a discussion on Monday.
Bajwa said the Supreme Court has reiterated that it is the duty of the Governor to "act judiciously" while exercising their powers under Article 164 at the time of appointing the Chief Minister.
When BJP and alliance partners gave the letter of support to Governor Mridula Sinha, the Congress woke up from its slumber and got into action, Rane said.
The Goa government announced that its Budget will be presented on March 24, and the Governor's address will take place on 23.
Parrikar also rubbished allegations of horse trading levelled by the Congress against the BJP.
Congress leaders had accused the BJP of indulging in "unfair means" to cobble up the numbers to retain power in the coastal state.
Meanwhile, the Nationalist Congress Party issued show-cause notice to MLA Churchill Alemao for violating the party discipline by supporting the BJP during the floor test.
Goa Chief Minister Manohar Parrikar has won the trust vote in the Assembly with the support of smaller parties and independents.
Goa floor test: BJP's Manohar Parrikar has proved his majority in the Goa Assembly.
The BJP said it would not comment on the allegations as these were baseless.
Rane said he was "open" to quit his party if no action is taken against those responsible for the Congress' debacle.
Congress wins people by heart and not through money, he said.
"If Rahul Gandhi does not respond within two days, I will feel I am not the right fit in the party."
The court shot down Goa Congress Legislature Party (CLP) leader Chandrakant Kavlekar’s plea that the Governor should not have appointed Parrikar as Chief Minister in the first place.
Goa elections: Chandrakant Kavlekar, Leader of the Congress Legislature Party, complained that he was not called even once by the Governor to form the government.
The first criteria, the Sarkaria Commission recommended, was that the “party or combination of parties which commands the widest support in the Legislative Assembly should be called upon to form the government”.
Hours after the results were announced, Sardesai, 47, whose party won three of four constituencies it contested, became the state’s most talked-about man.
A bench led by Chief Justice of India J S Khehar accepted Kavlekar’s submission that a floor test was the most valid method of proving majority and said it must be done “at the earliest”.
“She should have invited us as per the convention and procedure of the Constitution and now this is inviting us in deference of the Supreme Court order,” Luizinho said.
Manohar Parrikar resigned from his post as Defence minister and is set to take oath as Goa Chief Minister. He will have to go through a trust vote as well to prove majority in Assembly on Thursday.
Goa: Congress has urged like-minded parties to come together and support it in forming a non-BJP government. Vijai Sardesai, however, has said there's no question of the Goa Forward backing out.
The apex court today refused to stay the swearing-in of Parrikar as Goa CM and ordered a floor test on March 16
The Congress last night made a representation to the Governor asking her to invite the party to form the government in the coastal state.
The Goa Governor has already invited the BJP-led alliance to form the government and the swearing-in is scheduled this evening
Goa government formation LIVE updates: Manohar Parrikar sworn-in as chief minister of Goa. Supreme Court gives him 48 hours to prove majorityu on floor of the House. Follow for the latest updates.
Plea says Congress single largest party after polls, should have been called first to stake claim for government.
On Monday, Parrikar set the stage for his return as Chief Minister by resigning as Union Defence Minister — Finance Minister Arun Jaitley will take additional charge of the portfolio.
Moily, however, made it clear that this “of course does not apply to our leadership.Rahul Gandhi or Sonia Gandhi”.
Sources said that Shah was “quite confident” that it would not be a tough job for former Goa Chief Minister Parrikar, who has strong ties with most politicians in the state, to help the party form the government.
Parrikar — a reluctant defence minister and called the “weekend Chief Minister”, whose famous remark of preferring fish curry to butter chicken was seen as indication of him wanting to return to Panaji — desperately needed those coconuts to return to power.
Goa Forward had secured three seats in the recently held assembly elections in the coastal state.
Parrikar resigned from his position as MoD ahead of the swearing-in ceremony.
Amit Shah, Rajnath Singh and cabinet colleague Nitin Gadkari would be present in the swearing-in ceremony at Raj Bhavan on Tuesday evening.