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.