The case highlighted a shift in policy in Italy, with the country's new eurosceptic government presenting a notably less accommodating approach to immigration after more than 600,000 people arrived by sea over the past five years.
Sea Watch refused last week to take 40 migrants rescued by the U.S. navy ship Trenton off Libya, fearing a fate similar to that of the Aquarius.
It marked the close of an exhausting ordeal that began when the charity rescue boat was barred from entering ports in Italy and Malta, leaving the migrants on board, including 100 children and seven pregnant women, stranded at sea.
Doctors Without Borders say the migrants come from 26 countries, mainly from Africa, but also Afghanistan, Bangladesh and Pakistan. A lot of them from Africa but others are from Afghanistan, Pakistan and Bangladesh, according to medical charity Médecins Sans Frontières.
The migrants have sparked a political storm about Europe's migration policy.
Italian news reports say a U.S. Navy ship will offload the 40 migrants its crew rescued earlier this week onto an Italian coast guard vessel that will bring them to shore.
In addition, a new populist government in Italy - one whose interior minister has vowed to deport tens of thousands of migrants as soon as he can - will make any compromises on European Union migration policy even more hard.
"I made the decision to not accept the migrants because it was not our responsibility and because we were obeying worldwide law", Muscat said.
The meeting between Macron and Conte at the Elysee Palace in Paris nearly did not take place after the president offered a harsh assessment of Italy's refusal to accept a private rescue ship carrying 629 migrants.
On Saturday, the Spanish government announced it would transfer to France any migrants arriving on the convoy who expressed a desire to reach that country.
Migrants disembark from the MV Aquarius, a search and rescue ship run in partnership between SOS Mediterranee and Medecins Sans Frontieres, after it arrived in Augusta on the island of Sicily, Italy. It took the convoy almost a week to travel the 930 miles between Sicily and Spain.
France made the offer via its ambassador to Spain, Deputy PM Carmen Calvo said in a statement.
Muscat said on Sunday that while on its way to Italy, the ship was asked by Italian authorities to anchor in worldwide waters - a request which breached global law and risked causing a humanitarian crisis.
One man was hopeful about the prospect of a new life in Spain, where the migrants can apply for asylum, along with the possibility of seeking asylum in France, providing they had a legitimate claim.
Police and Civil Guard staff hold a briefing next to the point of arrival of refugee vessel Aquarius and two other Italian ships expected tomorrow at the port of Valencia, Spain, June 16, 2018.
Applause could be heard coming from the ship as it docked.
"They should know that Italy no longer wants to be an accomplice in the business of illegal immigration and therefore they will have to aim for other, non-Italian, ports", Salvini said.
"As minister and as a father, I take this action for the benefit of all", he added. In a Facebook post Saturday, Salvini said Lifeline and the Seefuchs ship of the German group Sea-Eye couldn't dock in Italy.