"There is a degree of cynicism and irresponsibility in the Italian government's behaviour", said French President Emmanuel Macron through a spokesman.
But in order to do so, coastguard authorities said migrants will need to disembark Aquarius and be transferred onto Italian government-run ships.
The refusals by Italy and Malta, leaving the Aquarius unable to quickly bring the migrants to a safe port, dismayed others.
The new Spanish foreign minister said the decision to offer a docking port in the eastern city of Valencia had been a "personal and direct" move by the country's new prime minister, the Socialist Pedro Sanchez.
"It is our duty to help avoid a humanitarian catastrophe and offer a safe port to these people, to comply with our human rights obligations", Sanchez's office said.
But the deteriorating weather and the extremely cramped conditions on board the Aquarius - which was built to transport a maximum of 500 - could make that journey to Valencia perilous.
"We are talking about people", commission spokesman Margaritis Schinas told a news conference. "629 immigrants overseas the ship Aquarius headed for Spain, primary goal achieved!" The group includes seven pregnant women, 123 unaccompanied minors and people suffering from burns caused from a mixture of seawater and fuel.
The ship is named Aquarius and operated by Doctors Without Borders (MSF) and the rescue organization SOS Mediterranee.
Global law requires that rescue vessels not be subject to "undue delay" in their mission and are provided assistance from coastal states "as soon as practicable".
Japanese PM Shinzo Abe considers joining in diplomacy with North Korea
Security around Kim's hotel remained tight, suggesting the North Korean leader may return after the summit. "President Obama said that North Korea was our biggest and most unsafe problem".
Malta's Prime Minsiter Joseph Muscat continued to insist his country is acting according to worldwide law and that it is Italy's duty to take in the migrants, as Italy's coast guard coordinated the rescues off the coast of Libya.
"Italy is done bending over backwards and obeying - this time THERE IS SOMEONE WHO SAYS NO", he wrote on Twitter.
"Enough!" Salvini said Monday.
"The statements concerning (the humanitarian ship) Aquarius that come from France are surprising".
Rescue ships like the Aquarius pick migrants up from those unsafe boats.
Mayor Filippo Nogarin - a member of the Five Star Movement (M5S), the anti-establishment party now governing Italy in coalition with the nationalist League - had said that he was ready to "open the port of Livorno and take in the ship Aquarius with its cargo of 629 human lives".
Catalonia's leader Quim Torra said that his region was capable of hosting a total of 1,800 refugees.
Euronews correspondent Anelise Borges said the group consisted of 90 migrants, with more than 400 people remaining to be transferred according to an initial plan.
Under a European Union agreement, the country where asylum-seekers arrive and are identified must care for them until their asylum requests are decided, a process that can take a couple of years.