During the informal summit, EU member states - apart from Britain - were expected to discuss at the how to respond to the Brexit vote and consider future development of the EU itself.
According to Slovak Prime Minister Robert Fico, the main issues to be discussed in Bratislava will be migration and its economic dimension and the EU's future following Brexit.
But, on Thursday, Tusk appeared to distance himself from this, saying he did not want Brexit to be formally discussed at the meeting on 16 September.
In a statement, he said, "It is not our intention to talk about the UK in Bratislava or our negotiation strategy vis-à-vis the UK. Our position is crystal clear: there will be no negotiations without notification.
"This principle is enshrined in our treaties. And it is there for a reason: to protect the interests of the members of the Union that want to stay together, not the one which decides to leave.
"Therefore, we shall not give up on this principle."
Tusk, a former Polish Prime Minister, said, "We need to come up with a common diagnosis of the EU after the vote in the UK. We will not solve all the EU's problems overnight, but what we need to do there is to establish common objectives.
"This will allow us to rebuild a sense of political unity in the months to come, also ahead of our meeting in Rome in March 2017, 60 years after the founding of our community.
"We need that sense of political unity much more today, with the world around Europe bringing more threats than opportunities, than we did in the sunny days."
He made his comments after meeting Luxembourg Prime Minister Xavier Bettel in the Grand Duchy on Thursday.
In comments that may appear to be a swipe at Europe's political elite, he was also later reported as saying, "We must help people to restore faith in the fact that the EU should serve them, guarantee their protection and share their emotions.
"All too often today, the European elites seem to be detached from reality. We need to discuss what Brexit means politically for the EU. We need to talk about ourselves, the EU, that sooner or later will be left by the UK.
"It is my deep conviction that our priority should be to take back full control in the field of internal security, and on our external borders. It is not an accident that this issue, together with migration, was at the forefront of the referendum campaign in the UK.
"But the sense of insecurity and destabilisation, in many aspects, including the economic one, is present also on the European continent."
Meanwhile, Frans Timmermans, European Commission First Vice-President, has said Britain's political leaders should "get their act together" and tell the EU what they want from their divorce.
Timmermans said Brexit was an "existential issue" for the rest of the bloc, but the Union had "enormous possibilities" to thrive.
The Dutch official said, "I follow closely what is happening in the UK and the political discussions in the UK, and perhaps they should first get their act together and tell us what they really want out of this."
"It's not the 27 other member states who asked the UK to leave. On the contrary, they pleaded with the UK to stay," he added in an interview to mark Thursday's publication in French of Fraternité, a book on Europe.