The French president's office said France didn't want to "start a precedent" that would allow some European countries to breach worldwide laws and rely on other EU member states to take in migrants.
A government source said Conte still did not see the right "conditions" for a Friday meeting with Macron.
Interior minister Matteo Salvini, who is deputy premier and leader of the anti-immigrant League Party, called on Mr Macron to apologise and said he was not prepared to take criticism from a country which regularly stopped migrants on their shared border.
"If an official apology doesn't arrive, prime minister Conte would be right not to go to France", Salvini told reporters after demanding that France take in more migrants.
Al Jazeera's Jonah Hull reports from Rome.
"It is shameful. As a European, I felt shame - shame - that there was a boat - there is a boat - in the Mediterranean, and for several days nobody wanted to take these people", Grandi told Reuters.
Macron said Rome had acted "with cynicism and irresponibility" by turning away the migrant ship Aquarius. Foreign Ministry spokeswoman Agnes von der Muhll said France was "fully aware of the burden that the migration pressure is placing on Italy" and that it was committed to cooperating with Rome on immigration.
Lashing out at the French government, Salvini said France had taken in only a fraction of the 9,816 migrants it had pledged to accept under a 2015 European Union relocation plan.
The vessel, carrying 629 migrants, is now heading to Valencia, Spain, escorted by two Italian ships.
"So I ask President Macron to pass from words to action and tomorrow morning, welcome the 9,000 France promised to welcome as a sign of concrete generosity and not just words", Salvini said.
Under EU rules, migrants must apply for asylum in the European nation where they first arrive.
The chancellor said migration represented a significant challenge for Europe when it was in a "very fragile situation".
It would also represent a complete reversal of the open-doors migrant policy she adopted in 2015, which saw more than 1 million migrants enter Germany and is seen as having fuelled a sharp rise in support for the rightwing Alternative for Germany (AfD) party and caused deep rifts within her conservative bloc.
As EU politicians have dithered or traded blame, refugees have continued to arrive.
Italian maritime vessels that have picked up Europe-bound migrants in the Mediterranean still are entering the ports; on Wednesday, an Italian coast guard vessel docked in Catania, Sicily with 932 migrants on board. German aid group Sea-Watch said the migrants were picked up after their rubber dinghy sank and the Navy also had recovered the bodies of 12 people.
However, the Sea Watch is still waiting to pick them up after asking for a written guarantee that it will be allowed to dock in an Italian port.