Interior Minister Matteo Salvini then suggested that Conte call off the meeting with Macron and demanded that France apologize.
He also said that between January and May, France had sent back to Italy 10,249 migrants who had crossed their common border. The Aquarius was heading for Spain on Wednesday.
"As a European I feel ashamed that there is a boat roaming around the sea", UN High Commissioner for Refugees Filippo Grandi told reporters in Geneva.
The case touched on one of the main faultlines in European politics - how to share the responsibility of handling migrants trying to get into the bloc from war zones and poor countries, largely across Africa and the Middle East.
"Europe must stick together", she said, acknowledging Italy's concern that it had taken in many migrants but fellow European Union countries were not showing sufficient solidarity.
Salvini's League scored its best result in the March national election, partly on pledges to deport hundreds of thousands of migrants and halt the flow of newcomers, and has formed a coalition with the anti-establishment 5-Star Movement. "Italy is right to say that this has to be shared more widely". The border crossing point at Ventimiglia has been the scene of protests and desperation for years as France has refused to let in migrants, many of whom are seeking to reach family in France or Germany.
The French government was more conciliatory Wednesday, when the Foreign Ministry said France recognized the burden Italy has carried.
Both leaders discussed the situation of the Aquarius ship and agreed on France and Italy's commitment to organize emergency aid under humanitarian rules to assist people in danger, according to the French presidency.More news: Meghan Markle has been submitted for Emmy consideration for 'Suits'
Italy responded to French criticism by summoning its ambassador.
A government source said Conte still did not see the right "conditions" for a Friday meeting with Macron.
Sophie Beau said Wednesday in a news conference in Marseille, France, "it's a relief for everyone, our teams and of course above all for the survivors to know that they are finally allowed to head to a safe port in Europe".
After meeting in Berlin, Austrian Chancellor Sebastian Kurz and German Interior Minister Horst Seehofer echoed Salvini.
"We believe an axis of the willing is needed to fight illegal migration". She also describes the fear that the migrants were experiencing as "so present, so palpable".
As EU politicians have dithered or traded blame, refugees have continued to arrive.
Meanwhile, an Italian coastguard ship carrying more than 900 migrants was allowed to dock in Sicily on Wednesday.
The port of Valencia is some 1,500 kilometres (930 miles) from where the vessel had been on standby.